Codebook design method for multiple input multiple output system and method for using the codebook

ABSTRACT

A multiple input multiple output (MIMO) communication method using a codebook is provided. The MIMO communication method may use one or more codebooks and the codebooks may change according to a transmission rank, a channel state of a user terminal, and/or a number of feedback bits. The one or more codebooks may be adaptively updated according to a time correlation coefficient of a channel.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit under 35 U.S.C. §119(e) of a U.S.Provisional Application No. 61/076,179, filed on Jun. 27, 2008 in theUnited States Patent and Trade mark Office, and the benefit under 35U.S.C. §119(a) of a Korean Patent Application No. 10-2008-0080076, filedon Aug. 14, 2008, and a Korean Patent Application No. 10-2009-0017040,filed on Feb. 27, 2009, in the Korean Intellectual Property Office, theentire disclosures of which are incorporated herein by reference for allpurposes.

BACKGROUND

1. Field

The following description relates to a codebook that is used in amultiple input multiple output (MIMO) communication system.

2. Description of the Related Art

A number of researches are being conducted to provide various types ofmultimedia services such as voice services and to support high qualityand high speed of data transmission in a wireless communicationenvironment. Technologies associated with a multiple input multipleoutput (MIMO) communication system using multiple channels are in rapiddevelopment.

In a MIMO communication system, a base station and terminals may use acodebook in order to appropriately cope with a channel environment. Aparticular space may be quantized into a plurality of codewords. Theplurality of codewords that is generated by quantizing the particularspace may be stored in the base station and the terminals. Each of thecodewords may be a vector or a matrix according to the dimension of achannel matrix.

For example, each of the terminals may select a matrix or a vectorcorresponding to channel information from matrices or vectors includedin a codebook, based on a channel that is formed between the basestation and each of the terminals. The base station may also receive theselected matrix or vector based on the codebook to thereby recognize thechannel information. The selected matrix or vector may be used where thebase station performs beamforming or transmits a transmission signal viamultiple antennas.

Accordingly, there is a need for a well-designed codebook in order toimprove a performance of a MIMO communication system.

SUMMARY

In one general aspect, a multiple input multiple output (MIMO)communication method includes storing a codebook that includes at leastone matrix or vector, determining a transmission rank corresponding to anumber of data streams, generating, using the at least one matrix orvector that is included in the codebook, a precoding matrix according toa channel state of at least one user terminal and the transmission rank,and precoding the data streams using the precoding matrix.

In another general aspect, a MIMO communication method includes storinga codebook that includes at least one matrix or vector, updating thecodebook according to a time correlation coefficient (ρ) of a channelthat is formed between at least one user terminal and a base station,generating a precoding matrix using at least one matrix or vectorincluded in the updated codebook, and precoding the data streams usingthe precoding matrix.

The updating of the precoding matrix may include updating a previousprecoding matrix to a new precoding matrix.

The updating of the codebook may include updating the codebook using thefollowing Equation,

${\overset{\sim}{\Theta}}_{i} = {\underset{{\overset{\sim}{\Theta}}_{i}}{argmin}{{{{\Psi_{i}( {\rho,\Theta_{i}} )} - {\overset{\sim}{\Theta}}_{i}}}_{F}.}}$Here, Ψ_(i)(ρ, Θ_(i))=ρI+√{square root over (1−ρ²)}Θ_(i), i=1, 2, 3, . .. , 2^(B), B denotes a number of feedback bits, Θ_(i) denotes a unitarymatrix or a diagonal matrix as an i^(th) element of the codebook{θ}={Θ₁, . . . , Θ₂ _(B) }, I denotes an identity matrix, ∥x∥_(F)denotes a Frobenious norm for x, and {tilde over (Θ)}_(i) denotes ani^(th) element of the updated codebook {{tilde over (θ)}}={{tilde over(Θ)}₁, . . . , {tilde over (Θ)}₂ _(B) }.

Where a singular value decomposition (SVD) is performed forΨ_(i)(ρ,Θ_(i)), Ψ_(i)(ρ,Θ_(i)) may be expressed as Φ_(i)Λ_(i)B_(i)*.B*_(i) denotes a conjugate matrix of B_(i). The above equation

${\overset{\sim}{\Theta}}_{i} = {\underset{{\overset{\sim}{\Theta}}_{i}}{argmin}{{{\Psi_{i}( {\rho,\Theta_{i}} )} - {\overset{\sim}{\Theta}}_{i}}}_{F}}$may be optimized to {tilde over (Θ)}_(i)=Φ_(i)B_(i)*. The codebook maybe updated using {tilde over (Θ)}_(i)=Φ_(i)B_(i)*.

The updating of the codebook may include updating the codebook using thefollowing Equation,{tilde over(Θ)}_(i)=[Ψ_(i)(ρ,Θ_(i))*Ψ_(i)(ρ,Θ_(i))]^(−1/2)Ψ_(i)(ρ,Θ_(i)).Here, {tilde over (Θ)}_(i) denotes an i^(th) element of the updatedcodebook {{tilde over (θ)}}={{tilde over (Θ)}₁, . . . , {tilde over(Θ)}₂ _(B) }, i=1, 2, 3, . . . 2^(B), Ψ_(i)(ρ,Θ_(i))=ρI+√{square rootover (1−ρ²)}Θ_(i), Θ_(i) denotes a unitary matrix or a diagonal matrixas an i^(th) element of the codebook {θ}={Θ₁, . . . , Θ₂ _(B) }, and Idenotes an identity matrix.

In still another general aspect, a base station for a single usermultiple input multiple output (MIMO) communication system, includes amemory where a codebook including 4×1 codeword matrices C_(1,1),C_(2,1), C_(3,1), C_(4,1), C_(5,1), C_(6,1), C_(7,1), C_(8,1), C_(9,1),C_(10,1), C_(11,1), C_(12,1), C_(13,1), C_(14,1), C_(15,1), and C_(16,1)is stored, and a precoder to precode a data stream to be transmittedusing the codebook, wherein the codeword matrices are defined by thefollowing table:

C_(1,1) =   0.5000 C_(2,1) = −0.5000 C_(3,1) = −0.5000 C_(4,1) =  0.5000 −0.5000 −0.5000   0.5000 0 − 0.5000i   0.5000   0.5000   0.5000  0.5000 −0.5000   0.5000 −0.5000 0 − 0.5000i C_(5,1) = −0.5000 C_(6,1)= −0.5000 C_(7,1) =   0.5000 C_(8,1) =   0.5000 0 − 0.5000i 0 + 0.5000i  0.5000 0 + 0.5000i   0.5000   0.5000   0.5000   0.5000 0 + 0.5000i 0 −0.5000i   0.5000 0 + 0.5000i C_(9,1) =   0.5000 C_(10,1) =   0.5000C_(11,1) =   0.5000 C_(12,1) =   0.5000   0.5000 0 + 0.5000i −0.5000 0 −0.5000i   0.5000 −0.5000   0.5000 −0.5000 −0.5000 0 + 0.5000i   0.5000 0− 0.5000i C_(13,1) =   0.5000 C_(14,1) =   0.5000 C_(15,1) =   0.5000C_(16,1) =   0.5000 0.3536 + 0.3536i −0.3536 + 0.3536i −0.3536 − 0.3536i0.3536 − 0.3536i 0 + 0.5000i 0 − 0.5000i 0 + 0.5000i 0 − 0.5000i−0.3536 + 0.3536i 0.3536 + 0.3536i 0.3536 − 0.3536i −0.3536 − 0.3536i

The precoder may calculate a precoding matrix based on at least onecodeword matrix among the codeword matrices, and precodes the datastream using the precoding matrix.

The base station may further include an information receiver to receivefeedback information from a terminal, wherein the precoder may precodethe data stream using the feedback information and the codebook.

The precoder may calculate a precoding matrix based on a codeword matrixcorresponding to the feedback information among the codeword matrices,and precode the data stream using the precoding matrix.

The feedback information may include index information of a codewordmatrix preferred by the terminal among the codeword matrices.

The base station may further include four transmit antennas, wherein thecodebook may be used for transmission rank 1.

In still another general aspect, a base station for a multi-user MIMOcommunication system, includes a memory where a codebook including 4×1codeword matrices C_(1,1), C_(2,1), C_(3,1), C_(4,1), C_(5,1), C_(6,1),C_(7,1), C_(8,1), C_(9,1), C_(10,1), C_(11,1), C_(12,1), C_(13,1),C_(14,1), C_(15,1), and C_(16,1) is stored, and a precoder to precode atleast one data stream to be transmitted using the codebook, wherein thecodeword matrices are defined by the following table:

C_(1,1) =  0.5000 C_(2,1) = −0.5000 C_(3,1) = −0.5000 C_(4,1) =  0.5000−0.5000 −0.5000  0.5000 0 − 0.5000i  0.5000  0.5000  0.5000  0.5000−0.5000  0.5000 −0.5000 0 − 0.5000i C_(5,1) = −0.5000 C_(6,1) = −0.5000C_(7,1) =  0.5000 C_(8,1) =  0.5000 0 − 0.5000i 0 + 0.5000i  0.5000 0 +0.5000i  0.5000  0.5000  0.5000  0.5000 0 + 0.5000i 0 − 0.5000i  0.50000 + 0.5000i C_(9,1) =  0.5000 C_(10,1) =  0.5000 C_(11,1) =  0.5000C_(12,1) =  0.5000  0.5000 0 + 0.5000i −0.5000 0 − 0.5000i  0.5000−0.5000  0.5000 −0.5000 −0.5000 0 + 0.5000i  0.5000 0 − 0.5000i C_(13,1)=  0.5000 C_(14,1) = 0 .5000 C_(15,1) =  0.5000 C_(16,1) =  0.50000.3536 + 0.3536i −0.3536 + 0.3536i −0.3536 − 0.3536i 0.3536 − 0.3536i0 + 0.5000i 0 − 0.5000i 0 + 0.5000i 0 − 0.5000i −0.3536 + 0.3536i0.3536 + 0.3536i 0.3536 − 0.3536i −0.3536 − 0.3536i

The base station may further include an information receiver to receivefeedback information from at least two terminals, wherein the precodermay precode the at least one data stream using at least one of thefeedback information received from the at least two terminals, and thecodebook.

In still another general aspect, a terminal for a MIMO communicationsystem, includes a memory where a codebook including 4×1 codewordmatrices C_(1,1), C_(2,1), C_(3,1), C_(4,1), C_(5,1), C_(6,1), C_(7,1),C_(8,1), C_(9,1), C_(10,1), C_(11,1), C_(12,1), C_(13,1), C_(14,1),C_(15,1), and C_(16,1) is stored, and a feedback unit to feed back, to abase station, feedback information associated with a preferred codewordmatrix among the codeword matrices, wherein the codeword matrices aredefined by the following table:

C_(1,1) =  0.5000 C_(2,1) = −0.5000 C_(3,1) = −0.5000 C_(4,1) =  0.5000−0.5000 −0.5000  0.5000 0 − 0.5000i  0.5000  0.5000  0.5000  0.5000−0.5000  0.5000 −0.5000 0 − 0.5000i C_(5,1) = −0.5000 C_(6,1) = −0.5000C_(7,1) =  0.5000 C_(8,1) =  0.5000 0 − 0.5000i 0 + 0.5000i  0.5000 0 +0.5000i  0.5000  0.5000  0.5000  0.5000 0 + 0.5000i 0 − 0.5000i  0.50000 + 0.5000i C_(9,1) =  0.5000 C_(10,1) =  0.5000 C_(11,1) =  0.5000C_(12,1) =  0.5000  0.5000 0 + 0.5000i −0.5000 0 − 0.5000i  0.5000−0.5000  0.5000 −0.5000 −0.5000 0 + 0.5000i  0.5000 0 − 0.5000i C_(13,1)=  0.5000 C_(14,1) =  0.5000 C_(15,1) =  0.5000 C_(16,1) =  0.50000.3536 + 0.3536i −0.3536 + 0.3536i −0.3536 − 0.3536i 0.3536 − 0.3536i0 + 0.5000i 0 − 0.5000i 0 + 0.5000i 0 − 0.5000i −0.3536 + 0.3536i0.3536 + 0.3536i 0.3536 − 0.3536i −0.3536 − 0.3536i

The terminal may further include a channel estimation unit to estimate achannel between the base station and the terminal, wherein the feedbackunit may feed back, to the base station, the feedback informationdetermined based on the estimated channel.

In still another general aspect, a storage medium is provided where acodebook used by a base station and at least one terminal of a MIMOcommunication system is stored, wherein the codebook includes codewordmatrices C_(1,1), C_(2,1), C_(3,1), C_(4,1), C_(5,1), C_(6,1), C_(7,1),C_(8,1), C_(9,1), C_(10,1), C_(11,1), C_(12,1), C_(13,1), C_(14,1),C_(15,1), and C_(16,1), and the codeword matrices are defined by thefollowing table:

C_(1,1) =  0.5000 C_(2,1) = −0.5000 C_(3,1) = −0.5000 C_(4,1) =  0.5000−0.5000 −0.5000  0.5000 0 − 0.5000i  0.5000  0.5000  0.5000  0.5000−0.5000  0.5000 −0.5000 0 − 0.5000i C_(5,1) = −0.5000 C_(6,1) = −0.5000C_(7,1) =  0.5000 C_(8,1) =  0.5000 0 − 0.5000i 0 + 0.5000i  0.5000 0 +0.5000i  0.5000  0.5000  0.5000  0.5000 0 + 0.5000i 0 − 0.5000i  0.50000 + 0.5000i C_(9,1) =  0.5000 C_(10,1) =  0.5000 C_(11,1) =  0.5000C_(12,1) =  0.5000  0.5000 0 + 0.5000i −0.5000 0 − 0.5000i  0.5000−0.5000  0.5000 −0.5000 −0.5000 0 + 0.5000i  0.5000 0 − 0.5000i C_(13,1)=  0.5000 C_(14,1) =  0.5000 C_(15,1) =  0.5000 C_(16,1) =  0.50000.3536 + 0.3536i −0.3536 + 0.3536i −0.3536 − 0.3536i 0.3536 − 0.3536i0 + 0.5000i 0 − 0.5000i 0 + 0.5000i 0 − 0.5000i −0.3536 + 0.3536i0.3536 + 0.3536i 0.3536 − 0.3536i −0.3536 − 0.3536i

In still another general aspect, a precoding method of a base stationfor a single user MIMO communication system, includes accessing a memorywhere a codebook including 4×1 codeword matrices C_(1,1), C_(2,1),C_(3,1), C_(4,1), C_(5,1), C_(6,1), C_(7,1), C_(8,1), C_(9,1), C_(10,1),C_(11,1), C_(12,1), C_(13,1), C_(14,1), C_(15,1), and C_(16,1) isstored, and precoding a data stream to be transmitted using thecodebook, wherein the codeword matrices are defined by the followingtable:

C_(1,1) =  0.5000 C_(2,1) = −0.5000 C_(3,1) = −0.5000 C_(4,1) =  0.5000−0.5000 −0.5000  0.5000 0 − 0.5000i  0.5000  0.5000  0.5000  0.5000−0.5000  0.5000 −0.5000 0 − 0.5000i C_(5,1) = −0.5000 C_(6,1) = −0.5000C_(7,1) =  0.5000 C_(8,1) =  0.5000 0 − 0.5000i 0 + 0.5000i  0.5000 0 +0.5000i  0.5000  0.5000  0.5000  0.5000 0 + 0.5000i 0 − 0.5000i  0.50000 + 0.5000i C_(9,1) =  0.5000 C_(10,1) =  0.5000 C_(11,1) =  0.5000C_(12,1) =  0.5000  0.5000 0 + 0.5000i −0.5000 0 − 0.5000i  0.5000−0.5000  0.5000 −0.5000 −0.5000 0 + 0.5000i  0.5000 0 − 0.5000i C_(13,1)=  0.5000 C_(14,1) =  0.5000 C_(15,1) =  0.5000 C_(16,1) =  0.50000.3536 + 0.3536i −0.3536 + 0.3536i −0.3536 − 0.3536i 0.3536 − 0.3536i0 + 0.5000i 0 − 0.5000i 0 + 0.5000i 0 − 0.5000i −0.3536 + 0.3536i0.3536 + 0.3536i 0.3536 − 0.3536i −0.3536 − 0.3536i

The precoding may include calculating a precoding matrix based on atleast one codeword matrix among the codeword matrices, and precoding thedata stream using the precoding matrix.

The method may further include receiving feedback information from aterminal, wherein the precoding may include precoding the data streamusing the feedback information and the codebook.

In still another general aspect, a precoding method of a base stationfor a multi-user MIMO communication system, includes accessing a memorywhere a codebook including 4×1 codeword matrices C_(1,1), C_(2,1),C_(3,1), C_(4,1), C_(5,1), C_(6,1), C_(7,1), C_(8,1), C_(9,1), C_(10,0),C_(11,1), C_(12,1), C_(13,1), C_(14,1), C_(15,1), and C_(16,1) isstored, and precoding at least one data stream to be transmitted usingthe codebook, wherein the codeword matrices are defined by the followingtable:

C_(1,1) =  0.5000 C_(2,1) = −0.5000 C_(3,1) = −0.5000 C_(4,1) =  0.5000−0.5000 −0.5000  0.5000 0 − 0.5000i  0.5000  0.5000  0.5000  0.5000−0.5000  0.5000 −0.5000 0 − 0.5000i C_(5,1) = −0.5000 C_(6,1) = −0.5000C_(7,1) =  0.5000 C_(8,1) =  0.5000 0 − 0.5000i 0 + 0.5000i  0.5000 0 +0.5000i  0.5000  0.5000  0.5000  0.5000 0 + 0.5000i 0 − 0.5000i  0.50000 + 0.5000i C_(9,1) =  0.5000 C_(10,1) =  0.5000 C_(11,1) =  0.5000C_(12,1) =  0.5000  0.5000 0 + 0.5000i −0.5000 0 − 0.5000i  0.5000−0.5000  0.5000 −0.5000 −0.5000 0 + 0.5000i  0.5000 0 − 0.5000i C_(13,1)=  0.5000 C_(14,1) =  0.5000 C_(15,1) =  0.5000 C_(16,1) =  0.50000.3536 + 0.3536i −0.3536 + 0.3536i −0.3536 − 0.3536i 0.3536 − 0.3536i0 + 0.5000i 0 − 0.5000i 0 + 0.5000i 0 − 0.5000i −0.3536 + 0.3536i0.3536 + 0.3536i 0.3536 − 0.3536i −0.3536 − 0.3536i

In still another general aspect, an operating method of a terminal for aMIMO communication system, includes accessing a memory where a codebookincluding 4×1 codeword matrices C_(1,1), C_(2,1), C_(3,1), C_(4,1),C_(5,1), C_(6,1), C_(7,1), C_(8,1), C_(9,1), C_(10,1), C_(11,1),C_(12,1), C_(13,1), C_(14,1), C_(15,1), and C_(16,1) is stored, andfeeding back, to a base station, feedback information associated with apreferred codeword matrix among the codeword matrices, wherein thecodeword matrices are defined by the following table:

C_(1,1) = 0.5000 C_(2,1) = −0.5000 C_(3,1) = −0.5000 C_(4,1) = 0.5000−0.5000 −0.5000 0.5000 0 − 0.5000i 0.5000 0.5000 0.5000 0.5000 −0.50000.5000 −0.5000 0 − 0.5000i C_(5,1) = −0.5000 C_(6,1) = −0.5000 C_(7,1) =0.5000 C_(8,1) = 0.5000 0 − 0.5000i 0 + 0.5000i 0.5000 0 + 0.5000i0.5000 0.5000 0.5000 0.5000 0 + 0.5000i 0 − 0.5000i 0.5000 0 + 0.5000iC_(9,1) = 0.5000 C_(10,1) = 0.5000 C_(11,1) = 0.5000 C_(12,1) = 0.50000.5000 0 + 0.5000i −0.5000 0 − 0.5000i 0.5000 −0.5000 0.5000 −0.5000−0.5000 0 + 0.5000i 0.5000 0 − 0.5000i C_(13,1) = 0.5000 C_(14,1) =0.5000 C_(15,1) = 0.5000 C_(16,1) = 0.5000 0.3536 + 0.3536i −0.3536 +0.3536i −0.3536 − 0.3536i 0.3536 − 0.3536i 0 + 0.5000i 0 − 0.5000i 0 +0.5000i 0 − 0.5000i −0.3536 + 0.3536i 0.3536 + 0.3536i 0.3536 − 0.3536i−0.3536 − 0.3536i

The method may further include estimating a channel between the basestation and the terminal, and generating the feedback information basedon the estimated channel.

Other features will become apparent to those skilled in the art from thefollowing detailed description, which, taken in conjunction with theattached drawings, discloses exemplary embodiments.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a diagram illustrating a multiple input multiple output (MIMO)communication system according to an exemplary embodiment.

FIG. 2 is a block diagram illustrating a configuration of a base stationaccording to an exemplary embodiment.

FIG. 3 is a flowchart illustrating a MIMO communication method accordingto an exemplary embodiment.

FIG. 4 is a flowchart illustrating a MIMO communication method accordingto another exemplary embodiment.

Throughout the drawings and the detailed description, unless otherwisedescribed, the same drawing reference numerals will be understood torefer to the same elements, features, and structures. The elements maybe exaggerated for clarity and convenience.

DETAILED DESCRIPTION

The following detailed description is provided to assist the reader ingaining a comprehensive understanding of the methods, apparatuses and/orsystems described herein. Accordingly, various changes, modifications,and equivalents of the systems, apparatuses and/or methods describedherein will be suggested to those of ordinary skill in the art. Also,description of well-known functions and constructions are omitted toincrease clarity and conciseness.

Hereinafter, exemplary embodiments will be described in detail withreference to the accompanying drawings.

FIG. 1 illustrates a multiple input multiple output (MIMO) communicationsystem according to an exemplary embodiment. The MIMO communicationsystem may be a closed loop MIMO communication system.

Referring to FIG. 1, the MIMO communication system includes a basestation 110 and a plurality of users (user 1, user 2, user n_(u)) 120,130, and 140. While FIG. 1 shows an example of a multi-user MIMOcommunication system, it is understood that the disclosed systems,apparatuses and/or methods may be applicable to a single user MIMOcommunication system.

Herein, the term “closed-loop” may indicate that the plurality of users(user 1, user 2, user n_(u)) 120, 130, and 140 may feedback, to the basestation 110, feedback data containing channel information and the basestation 110 may generate a transmission signal based on the feedbackdata. Also, a codebook according to exemplary embodiments to bedescribed later may be applicable to an open-loop MIMO communicationsystem as well as the closed-loop MIMO communication system.Accordingly, it is understood that exemplary embodiments are not limitedto the closed-loop MIMO communication system.

A plurality of antennas may be installed in the base station 110. Asingle antenna or a plurality of antennas may be installed in each ofthe users (user 1, user 2, user n_(u)) 120, 130, and 140. A channel maybe formed between the base station 110 and each of the users (user 1,user 2, user n_(u)) 120, 130, and 140. Signals may be transmitted andreceived via each formed channel.

The base station 110 may transmit a single data stream or at least twodata streams to the plurality of users (user 1, user 2, user n_(u)) 120,130, and 140. In this instance, the base station 110 may adopt a spatialdivision multiplex access (SDMA) scheme or SDM scheme. The base station110 may select a precoding matrix from matrices included in a codebookand generate a transmission signal using the selected precoding matrix.

For example, the base station 110 may transmit pilot signals to theplurality of users (user 1, user 2, user n_(u)) 120, 130, and 140 viadownlink channels, respectively. The pilot signals may be well known tothe base station 110 and the plurality of users (user 1, user 2, usern_(u)) 120, 130, and 140.

A terminal corresponding to each of the users (user 1, user 2, usern_(u)) 120, 130, and 140 may perform receiving a well-known signaltransmitted from the base station 110, estimating a channel that isformed between the base station 110 and each of the users (user 1, user2, user n_(u)) 120, 130, and 140 using a pilot signal, selecting atleast one matrix or vector from a codebook, and feeding back informationassociated with the selected at least one matrix or vector. The codebookmay be designed according to descriptions that will be made later withreference to FIGS. 2 through 4. The codebook may be updated according toa channel state.

Specifically, each of the users (user 1, user 2, user n_(u)) 120, 130,and 140 may estimate the channel formed between the base station 110 andeach of the users (user 1, user 2, user n_(u)) 120, 130, and 140 usingthe pilot signal. Each of the users (user 1, user 2, user n_(u)) 120,130, and 140 may select, as a preferred vector, any one vector fromvectors that are included in a pre-stored codebook, or may select, as apreferred matrix, any one matrix from matrices that are included in thecodebook.

For example, each of the users (user 1, user 2, user n_(u)) 120, 130,and 140 may select, as the preferred vector or the preferred matrix, anyone vector or any one matrix from 2^(B) vectors or 2^(B) matricesaccording to an achievable data transmission rate or asignal-to-interference and noise ratio (SINR). Here, B denotes a numberof feedback bits. Each of the users (user 1, user 2, user n_(u)) 120,130, and 140 may determine its own preferred transmission rank. Thetransmission rank may correspond to a number of data streams.

Each of the users (user 1, user 2, user n_(u)) 120, 130, and 140 mayfeedback, to the base station 110, information associated with theselected preferred vector or preferred matrix (hereinafter, referred toas channel information). The channel information used herein may includechannel state information, channel quality information, or channeldirection information.

The base station 110 may receive channel information of each of theusers (user 1, user 2, user n_(u)) 120, 130, and 140 to therebydetermine a precoding matrix. The base station 110 may select a portionor all of the users (user 1, user 2, user n_(u)) 120, 130, and 140according to various types of selection algorithms such as asemi-orthogonal user selection (SUS) algorithm, a greedy user selection(GUS) algorithm, and the like.

The same codebook as the codebook that is stored in the plurality ofusers (user 1, user 2, user n_(u)) 120, 130, and 140 may be pre-storedin the base station 110. The base station 110 may determine theprecoding matrix based on matrices included in the pre-stored codebookusing the channel information that is fed back from the plurality ofusers (user 1, user 2, user n_(u)) 120, 130, and 140. The base station110 may determine the precoding matrix to maximize a total datatransmission rate, that is, a sum rate.

The base station 110 may precode data streams S₁ and S_(N) based on thedetermined precoding matrix to thereby generate a transmission signal. Aprocess of generating the transmission signal by the base station 110may be referred to as “beamforming”.

A channel environment between the base station 110 and the plurality ofusers (user 1, user 2, user n_(u)) 120, 130, and 140 may be variable.Where the base station 110 and the plurality of users (user 1, user 2,user n_(u)) 120, 130, and 140 use a fixed codebook, it may be difficultto adaptively cope with the varying channel environment. Although itwill be described in detail later, the base station 110 and theplurality of users (user 1, user 2, user n_(u)) 120, 130, and 140 mayadaptively cope with the varying channel environment to thereby updatethe codebook.

The base station 110 may generate a new precoding matrix. In particular,the base station 110 may update a previous precoding matrix to the newprecoding matrix using the updated codebook.

FIG. 2 illustrates a configuration of a base station according to anexemplary embodiment.

Referring to FIG. 2, a base station according to an exemplary embodimentincludes a layer mapping unit 210, a MIMO encoding unit 220, a precoder230, and N_(t) physical antennas 240.

At least one codeword for at least one user may be mapped to at leastone layer. Where the dimension of codeword x is N_(C)×1, the layermapping unit 210 may map the codeword x to the at least one layer usinga matrix P with the dimension of N_(s)×N_(c). Here, N_(s) denotes anumber of layers or a number of effective antennas. Accordingly, it ispossible to acquire the following Equation 1:s=Px  (1).

The MIMO encoding unit 220 may perform space-time modulation for s usinga matrix function M with the dimension of N_(s)×N_(s). The MIMO encodingunit 220 may perform space-frequency block coding, spatial multiplexing,and the like, according to a transmission rank.

The precoder 230 may precode outputs, that is, data streams of the MIMOencoding unit 220 to thereby generate the transmission signal to betransmitted via the physical antennas 240. The dimension or the numberof outputs, that is, the data streams of the MIMO encoding unit 220 mayindicate the transmission rank. The precoder 230 may generate thetransmission signal using a precoding matrix U with the dimension ofN_(t)×N_(s). Accordingly, it is possible to acquire the followingEquation 2:z=UM(s)  (2).

Hereinafter, W denotes the precoding matrix and R denotes thetransmission rank or the number of effective antennas. Here, thedimension of the precoding matrix W is N_(t)×R. Where the MIMO encodingunit 220 uses spatial multiplexing, Z may be given by the followingEquation 3:

$\begin{matrix}{z = {\overset{\_}{WB} = {{\begin{bmatrix}u_{11} & u_{1\; R} \\\vdots & \vdots \\u_{{Nt}\; 1} & u_{NtR}\end{bmatrix}\begin{bmatrix}s_{1} \\\vdots \\s_{R}\end{bmatrix}}.}}} & (3)\end{matrix}$

Referring to the above Equation 3, the precoding matrix W may also bereferred to as a weighting matrix. The dimension of the precoding matrixW may be determined according to the transmission rank and the number ofphysical antennas. For example, where the number N_(t) of physicalantennas is four and the transmission rank is 2, the precoding matrix Wmay be given by the following Equation 4:

$\begin{matrix}{W = {\begin{bmatrix}W_{11} & W_{12} \\W_{21} & W_{22} \\W_{31} & W_{32} \\W_{41} & W_{42}\end{bmatrix}.}} & (4)\end{matrix}$

Codebook Properties

The codebook used in a closed-loop MIMO communication system or anopen-loop MIMO communication system may include a plurality of matricesor a plurality of vectors. A precoding matrix or a precoding vector maybe determined based on the plurality of matrices or the plurality ofvectors included in the codebook. Therefore, it is desirable to designthe codebook well.

1) Codebook used in a downlink of a single user MIMO communicationsystem where a number of physical antennas of a base station is two:

For example, where the number of physical antennas of the base stationis two, the codebook used in the single user MIMO communication systemaccording to an exemplary embodiment may be designed as given by thefollowing Equation 5:

$\begin{matrix}{{{W_{1} = {\frac{1}{\sqrt{2}}*\begin{bmatrix}1 & 1 \\1 & {- 1}\end{bmatrix}}},{W_{2} = {\frac{1}{\sqrt{2}}*\begin{bmatrix}1 & 1 \\j & {- j}\end{bmatrix}}},{W_{3} = {{\begin{bmatrix}1 & 0 \\0 & \frac{1 + j}{\sqrt{2}}\end{bmatrix}*W_{1}} = {\frac{1}{\sqrt{2}}\begin{bmatrix}1 & 1 \\\frac{1 + j}{\sqrt{2}} & \frac{{- 1} - j}{\sqrt{2}}\end{bmatrix}}}},{and}}{W_{4} = {{\begin{bmatrix}1 & 0 \\0 & \frac{1 + j}{\sqrt{2}}\end{bmatrix}*W_{2}} = {{\frac{1}{\sqrt{2}}\begin{bmatrix}1 & 1 \\\frac{{- 1} + j}{\sqrt{2}} & \frac{1 - j}{\sqrt{2}}\end{bmatrix}}.}}}} & (5)\end{matrix}$

In this instance, matrices or vectors included in the codebook for thesingle user MIMO communication system may be determined as given by thefollowing Table 1:

Transmit Codebook Transmission Transmission Index Rank 1 Rank 2 1C_(1,1) = W1(;, 1) C_(1,2) = W1(;, 1 2) 2 C_(2,1) = W1(;, 2) C_(2,2) =W2(;, 1 2) 3 C_(3,1) = W2(;, 1) C_(3,2) = W3(;, 1 2) 4 C_(4,1) = W2(;,2) C_(4,2) = W4(;, 1 2) 5 C_(5,1) = W3(;, 1) n/a 6 C_(6,1) = W3(;, 2)n/a 7 C_(7,1) = W4(;, 1) n/a 8 C_(8,1) = W4(;, 2) n/a

Referring to the above Table 1, W_(k)(;,n) denotes an n^(th) columnvector of W_(k), and W_(k)(;,n m) denotes a matrix that includes then^(th) column vector and the m^(th) column vector of W_(k). Where thetransmission rank is 1, the precoding matrix may be any one of W₁(;,1),W₁(;,2), W₂(;,1), W₂(;,2), W₃(;,1), W₃(;,2), W₄(;,1), and W₄(;,2). Wherethe transmission rank is 2, the precoding matrix may be any one ofW₁(;,1 2), W₂(;,1 2), W₃(;,1 2), and W₄(;,1 2).

2) Codebook used in a downlink of a multi-user MIMO communication systemperforming unitary precoding where a number of physical antennas of abase station is two:

For example, where the number of physical antennas of the base stationis two, the codebook used in the downlink of the multi-user MIMOcommunication system according to an exemplary embodiment may bedesigned as given by the following Equation 6:

$\begin{matrix}{{W_{1} = {\frac{1}{\sqrt{2}}*\begin{bmatrix}1 & 1 \\1 & {- 1}\end{bmatrix}}}{W_{2} = {\frac{1}{\sqrt{2}}*{\begin{bmatrix}1 & 1 \\j & {- j}\end{bmatrix}.}}}} & (6)\end{matrix}$

Vectors included in the codebook for the multi-user MIMO communicationsystem performing unitary precoding may be determined as given by thefollowing Table 2:

Codeword used at terminal to quantize the Transmit Codebook Indexchannel directions 1 W1(;, 1) 2 W1(;, 2) 3 W2(;, 1) 4 W2(;, 2)

The vectors used in the above Table 2 may be used at the terminal forchannel quantization. The matrices used at the base station forprecoding may be given by the above Equation 6.

Referring to the above Table 2, where the transmission rank is 1, theprecoding matrix may be constructed by appropriately combining W₁(;,1),W₁(;,2), W₂(;,1), and W₂(;,2). Here, W_(k)(;,n) denotes the n^(th)column vector of W_(k).

Where the multi-user MIMO communication system performs non-unitaryprecoding, the codebook used at the terminal for channel quantizationmay be the same as rank 1 codebook that is used in the single user MIMOcommunication system. Accordingly, where the multi-user MIMOcommunication system performs non-unitary precoding, the codebook mayinclude W₁(;,1), W₁(;,2), W₂(;,1), W₂(;,2), W₃(;,1), W₃(;,2), W₄(;,1),and W₄(;,2) included in the above Equation 5. The base station may useonly a subset of the codebook for this rank 1 single user MIMOcommunication system.

3) A first example of a codebook used in a downlink of a single userMIMO communication system or a multi-user MIMO communication systemwhere a number of physical antennas of a base station is four:

3-1) Codebook used in a downlink of a single user MIMO communicationsystem where a number of physical antennas of a base station is four:

For example, where the number of physical antennas of the base stationis four, the codebook used in the downlink of the single user MIMOcommunication system according to an exemplary embodiment may bedesigned as given by the following Equation 7:

$\begin{matrix}{{{W_{1} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix}1 & 1 & 0 & 0 \\1 & {- 1} & 0 & 0 \\0 & 0 & 1 & 1 \\0 & 0 & 1 & {- 1}\end{bmatrix}}},{W_{2} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix}1 & 1 & 0 & 0 \\j & {- j} & 0 & 0 \\0 & 0 & 1 & 1 \\0 & 0 & j & {- j}\end{bmatrix}}},{W_{3} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix}1 & 1 & 0 & 0 \\1 & {- 1} & 0 & 0 \\0 & 0 & 1 & 1 \\0 & 0 & j & {- j}\end{bmatrix}}},{W_{4} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix}1 & 1 & 0 & 0 \\j & {- j} & 0 & 0 \\0 & 0 & 1 & 1 \\0 & 0 & 1 & {- 1}\end{bmatrix}}},{W_{5} = {{{{diag}( {1,1,1,{- 1}} )}*D\; F\; T} = {0.5*\begin{bmatrix}1 & 1 & 1 & 1 \\1 & j & {- 1} & {- j} \\1 & {- 1} & 1 & {- 1} \\{- 1} & j & 1 & {- j}\end{bmatrix}}}},{and}}\begin{matrix}{W_{6} = {{{diag}( {1,\frac{( {1 + j} )}{\sqrt{2}},j,\frac{( {{- 1} + j} )}{\sqrt{2}}} )}*D\; F\; T}} \\{= {0.5*{\begin{bmatrix}1 & 1 & 1 & 1 \\\frac{( {1 + j} )}{\sqrt{2}} & \frac{( {{- 1} + j} )}{\sqrt{2}} & \frac{( {{- 1} - j} )}{\sqrt{2}} & \frac{( {1 - j} )}{\sqrt{2}} \\j & {- j} & j & {- j} \\\frac{( {{- 1} + j} )}{\sqrt{2}} & \frac{( {1 + j} )}{\sqrt{2}} & \frac{( {1 - j} )}{\sqrt{2}} & \frac{( {{- 1} - j} )}{\sqrt{2}}\end{bmatrix}.}}}\end{matrix}} & (7)\end{matrix}$

Here, a rotation matrix is

${U_{rot} = {\frac{1}{\sqrt{2}}\begin{bmatrix}1 & 0 & {- 1} & 0 \\0 & 1 & 0 & {- 1} \\1 & 0 & 1 & 0 \\0 & 1 & 0 & 1\end{bmatrix}}},$a quadrature phase shift keying (QPSK) discrete Fourier transform (DFT)matrix is

${{D\; F\; T} = {0.5*\begin{bmatrix}1 & 1 & 1 & 1 \\1 & j & {- 1} & {- j} \\1 & {- 1} & 1 & {- 1} \\1 & {- j} & {- 1} & j\end{bmatrix}}},$diag(a, b, c, d) is a 4×4 matrix, and diagonal elements of diag(a, b, c,d) are a, b, c, and d, and all the remaining elements are zero.

The matrices included in the codebook for the single user MIMOcommunication system may be determined according to the transmissionrank, as given by the following Table 3:

Transmit Codebook Transmission Transmission Transmission TransmissionIndex Rank 1 Rank 2 Rank 3 Rank 4 1 C_(1,1) = W1(;, 2) C_(1,2) = W1(;, 12) C_(1,3) = W1(;, 1 2 3) C_(1,4) = W1(;, 1 2 3 4) 2 C_(2,1) = W1(;, 3)C_(2,2) = W1(;, 1 3) C_(2,3) = W1(;, 1 2 4) C_(2,4) = W2(;, 1 2 3 4) 3C_(3,1) = W1(;, 4) C_(3,2) = W1(;, 1 4) C_(3,3) = W1(;, 1 3 4) C_(3,4) =W3(;, 1 2 3 4) 4 C_(4,1) = W2(;, 2) C_(4,2) = W1(;, 2 3) C_(4,3) = W1(;,2 3 4) C_(4,4) = W4(;, 1 2 3 4) 5 C_(5,1) = W2(;, 3) C_(5,2) = W1(;, 24) C_(5,3) = W2(;, 1 2 3) C_(5,4) = W5(;, 1 2 3 4) 6 C_(6,1) = W2(;, 4)C_(6,2) = W1(;, 3 4) C_(6,3) = W2(;, 1 2 4) C_(6,4) = W6(;, 1 2 3 4) 7C_(7,1) = W3(;, 1) C_(7,2) = W2(;, 1 3) C_(7,3) = W2(;, 1 3 4) n/a 8C_(8,1) = W4(;, 1) C_(8,2) = W2(;, 1 4) C_(8,3) = W2(;, 2 3 4) n/a 9C_(9,1) = W5(;, 1) C_(9,2) = W2(;, 2 3) C_(9,3) = W3(;, 1 2 3) n/a 10C_(10,1) = W5(;, 2) C_(10,2) = W2(;, 2 4) C_(10,3) = W3(;, 1 3 4) n/a 11C_(11,1) = W5(;, 3) C_(11,2) = W3(;, 1 3) C_(11,3) = W4(;, 1 2 3) n/a 12C_(12,1) = W5(;, 4) C_(12,2) = W3(;, 1 4) C_(12,3) = W4(;, 1 3 4) n/a 13C_(13,1) = W6(;, 1) C_(13,2) = W4(;, 1 3) C_(13,3) = W5(;, 1 2 3) n/a 14C_(14,1) = W6(;, 2) C_(14,2) = W4(;, 1 4) C_(14,3) = W5(;, 1 3 4) n/a 15C_(15,1) = W6(;, 3) C_(15,2) = W5(;, 1 3) C_(15,3) = W6(;, 1 2 4) n/a 16C_(16,1) = W6(;, 4) C_(16,2) = W6(;, 2 4) C_(16,3) = W6(;, 2 3 4) n/a

Referring to the above Table 3, where the transmission rank is 4, theprecoding matrix may be generated based on any one of W₁(;,1 2 3 4),W₂(;,1 2 3 4), W₃(;,1 2 3 4), W₄(;,1 2 3 4), W₅(;,1 2 3 4), and W₆(;,1 23 4). Here, W_(k)(;,n m o p) denotes a matrix that includes the n^(th)column vector, the m^(th) column vector, an o^(th) column vector, and ap^(th) column vector of W_(k).

Where the transmission rank is 3, the precoding matrix may be generatedbased on any one of W₁(;,1 2 3), W₁(;,1 2 4), W₁(;,1 3 4), W₁(;,2 3 4),W₂(;,1 2 3), W₂(;,1 2 4), W₂(;,1 3 4), W₂(;,2 3 4), W₃(;,1 2 3), W₃(;,13 4), W₄(;,1 2 3), W₄(;,1 3 4), W₅(;,1 2 3), W₅(;,1 3 4), W₆(;,1 2 4),and W₆(;,2 3 4). Here, W_(k)(;,n m o) denotes a matrix that includes then^(th) column vector, the column vector, and the o^(th) column vector ofW_(k).

Where the transmission rank is 2, the precoding matrix may be generatedbased on any one of W₁(;,1 2), W₁(;,1 3), W₁(;,1 4), W₁(;,2 3), W₁(;,24), W₁(;,3 4), W₂(;,1 3), W₂(;,1 4), W₂(;,2 3), W₂(;,2 4), W₃(;,1 3),W₃(;,1 4), W₄(;,1 3), W₄(;,1 4), W₅(;,1 3), and W₆(;,2 4). Here,W_(k)(;,n m) denotes a matrix that includes the n^(th) column vector andthe m^(th) column vector of W_(k).

Where the transmission rank is 1, the precoding matrix may be generatedbased on any one of W₁(;,2), W₁(;,3), W₁(;,4), W₂(;,2), W₂(;,3),W₂(;,4), W₃(;,1), W₄(;,1), W₅(;,1), W₅(;,2), W₅(;,3), W₅(;,4), W₆(;,1),W₆(;,2), W₆(;,3), and W₆(;,4). Here, W_(k)(;,n) denotes the n^(th)column vector of W_(k).

The codewords included in the above Table 3 may be expressed as follows:

$C_{1,1} = \begin{matrix}0.5000 \\{- 0.5000} \\0.5000 \\{- 0.5000}\end{matrix}$ $C_{2,1} = \begin{matrix}{- 0.5000} \\{- 0.5000} \\0.5000 \\0.5000\end{matrix}$ $C_{3,1} = \begin{matrix}{- 0.5000} \\0.5000 \\0.5000 \\{- 0.5000}\end{matrix}$ $C_{4,1} = \begin{matrix}0.5000 \\{0 - {0.5000i}} \\0.5000 \\{0 - {0.5000i}}\end{matrix}$

$C_{5,1} = \begin{matrix}{- 0.5000} \\{0 - {0.5000i}} \\0.5000 \\{0 + {0.5000i}}\end{matrix}$ $C_{6,1} = \begin{matrix}{- 0.5000} \\{0 + {0.5000i}} \\0.5000 \\{0 - {0.5000i}}\end{matrix}$ ${C_{7,1} = \begin{matrix}0.5000 \\0.5000 \\0.5000 \\0.5000\end{matrix}}\;$ $C_{8,1} = \begin{matrix}0.5000 \\{0 + {0.5000i}} \\0.5000 \\{0 + {0.5000i}}\end{matrix}$ $C_{9,1} = \begin{matrix}0.5000 \\0.5000 \\0.5000 \\{- 0.5000}\end{matrix}$ $C_{10,1} = \begin{matrix}0.5000 \\{0 + {0.5000i}} \\{- 0.5000} \\{0 + {0.5000i}}\end{matrix}$

$C_{11,1} = \begin{matrix}0.5000 \\{- 0.5000} \\0.5000 \\0.5000\end{matrix}$ $C_{12,1} = \begin{matrix}0.5000 \\{0 - {0.5000i}} \\{- 0.5000} \\{0 - {0.5000i}}\end{matrix}$ $C_{13,1} = \begin{matrix}0.5000 \\{0.3536 + {0.3536i}} \\{0 + {0.5000i}} \\{{- 0.3536} + {0.3536i}}\end{matrix}$ $C_{14,1} = \begin{matrix}0.5000 \\{{- 0.3536} + {0.3536i}} \\{0 - {0.5000i}} \\{0.3536 + {0.3536i}}\end{matrix}$ $C_{15,1} = \begin{matrix}0.5000 \\{{- 0.3536} - {0.3536i}} \\{0 + {0.5000i}} \\{0.3536 - {0.3536i}}\end{matrix}$ $C_{16,1} = \begin{matrix}0.5000 \\{0.3536 - {0.3536i}} \\{0 - {0.5000i}} \\{{- 0.3536} - {0.3536i}}\end{matrix}$

$C_{1,2} = \begin{matrix}0.5000 & 0.5000 \\0.5000 & {- 0.5000} \\0.5000 & 0.5000 \\0.5000 & {- 0.5000}\end{matrix}$ $C_{2,2} = \begin{matrix}0.5000 & {- 0.5000} \\0.5000 & {- 0.5000} \\0.5000 & 0.5000 \\0.5000 & 0.5000\end{matrix}$ $C_{3,2} = \begin{matrix}0.5000 & {- 0.5000} \\0.5000 & 0.5000 \\0.5000 & 0.5000 \\0.5000 & {- 0.5000}\end{matrix}$ $C_{4,2} = \begin{matrix}0.5000 & {- 0.5000} \\{- 0.5000} & {- 0.5000} \\0.5000 & 0.5000 \\{- 0.5000} & 0.5000\end{matrix}$ $C_{5,2} = \begin{matrix}0.5000 & {- 0.5000} \\{- 0.5000} & 0.5000 \\0.5000 & 0.5000 \\{- 0.5000} & {- 0.5000}\end{matrix}$ $C_{6,2} = \begin{matrix}{- 0.5000} & {- 0.5000} \\{- 0.5000} & 0.5000 \\0.5000 & 0.5000 \\0.5000 & {- 0.5000}\end{matrix}$

$C_{7,2} = \begin{matrix}0.5000 & {- 0.5000} \\{0 + {0.5000i}} & {0 - {0.5000i}} \\0.5000 & 0.5000 \\{0 + {0.5000i}} & {0 + {0.5000i}}\end{matrix}$ $C_{8,2} = \begin{matrix}0.5000 & {- 0.5000} \\{0 + {0.5000i}} & {0 + {0.5000i}} \\0.5000 & 0.5000 \\{0 + {0.5000i}} & {0 - {0.5000i}}\end{matrix}$ $C_{9,2} = \begin{matrix}0.5000 & {- 0.5000} \\{0 - {0.5000i}} & {0 - {0.5000i}} \\0.5000 & 0.5000 \\{0 - {0.5000i}} & {0 + {0.5000i}}\end{matrix}$ $C_{10,2} = \begin{matrix}0.5000 & {- 0.5000} \\{0 - {0.5000i}} & {0 + {0.5000i}} \\0.5000 & 0.5000 \\{0 - {0.5000i}} & {0 - {0.5000i}}\end{matrix}$ $C_{11,2} = \begin{matrix}0.5000 & 0.5000 \\0.5000 & {0 - {0.5000i}} \\0.5000 & 0.5000 \\0.5000 & {0 + {0.5000i}}\end{matrix}$ $C_{12,2} = \begin{matrix}0.5000 & {- 0.5000} \\0.5000 & {0 + {0.5000i}} \\0.5000 & 0.5000 \\0.5000 & {0 - {0.5000i}}\end{matrix}$

$C_{13,2} = \begin{matrix}0.5000 & {- 0.5000} \\{0 + {0.5000i}} & {- 0.5000} \\0.5000 & 0.5000 \\{0 + {0.5000i}} & 0.5000\end{matrix}$ $C_{14,2} = \begin{matrix}0.5000 & {- 0.5000} \\{0 + {0.5000i}} & 0.5000 \\0.5000 & 0.5000 \\{0 + {0.5000i}} & {- 0.5000}\end{matrix}$ $C_{15,2} = \begin{matrix}0.5000 & 0.5000 \\0.5000 & {- 0.5000} \\0.5000 & 0.5000 \\{- 0.5000} & 0.5000\end{matrix}$ $C_{16,2} = \begin{matrix}0.5000 & 0.5000 \\{{- 0.3536} + {0.3536i}} & {0.3536 - {0.3536i}} \\{0 - {0.5000i}} & {0 - {0.5000i}} \\{0.3536 + {0.3536i}} & {{- 0.3536} - {0.3536i}}\end{matrix}$ $C_{1,3} = \begin{matrix}0.5000 & 0.5000 & {- 0.5000} \\0.5000 & {- 0.5000} & {- 0.5000} \\0.5000 & 0.5000 & 0.5000 \\0.5000 & {- 0.5000} & 0.5000\end{matrix}$ $C_{2,3} = \begin{matrix}0.5000 & 0.5000 & {- 0.5000} \\0.5000 & {- 0.5000} & 0.5000 \\0.5000 & 0.5000 & 0.5000 \\0.5000 & {- 0.5000} & {- 0.5000}\end{matrix}$

$C_{3,3} = \begin{matrix}0.5000 & {- 0.5000} & {- 0.5000} \\0.5000 & {- 0.5000} & 0.5000 \\0.5000 & 0.5000 & 0.5000 \\0.5000 & 0.5000 & {- 0.5000}\end{matrix}$ $C_{4,3} = \begin{matrix}0.5000 & {- 0.5000} & {- 0.5000} \\{- 0.5000} & {- 0.5000} & 0.5000 \\0.5000 & 0.5000 & 0.5000 \\{- 0.5000} & 0.5000 & {- 0.5000}\end{matrix}$ $C_{5,3} = \begin{matrix}0.5000 & 0.5000 & {- 0.5000} \\{0 + {0.5000i}} & {0 - {0.5000i}} & {0 - {0.5000i}} \\0.5000 & 0.5000 & 0.5000 \\{0 + {0.5000i}} & {0 - {0.5000i}} & {0 + 0.5000}\end{matrix}$ $C_{6,3} = \begin{matrix}0.5000 & 0.5000 & {- 0.5000} \\{0 + {0.5000i}} & {0 - {0.5000i}} & {0 + {0.5000i}} \\0.5000 & 0.5000 & 0.5000 \\{0 + {0.5000i}} & {0 - {0.5000i}} & {0 - {0.5000i}}\end{matrix}$ $C_{7,3} = \begin{matrix}0.5000 & {- 0.5000} & {- 0.5000} \\{0 + {0.5000i}} & {0 - {0.5000i}} & {0 + {0.5000i}} \\0.5000 & 0.5000 & 0.5000 \\{0 + {0.5000i}} & {0 + {0.5000i}} & {0 - {0.5000i}}\end{matrix}$ $C_{8,3} = \begin{matrix}0.5000 & {- 0.5000} & {- 0.5000} \\{0 - {0.5000i}} & {0 - {0.5000i}} & {0 + {0.5000i}} \\0.5000 & 0.5000 & 0.5000 \\{0 - {0.5000i}} & {0 + {0.5000i}} & {0 - {0.5000i}}\end{matrix}$

$C_{9,3} = \begin{matrix}0.5000 & 0.5000 & {- 0.5000} \\0.5000 & {- 0.5000} & {0 - {0.5000\; i}} \\0.5000 & 0.5000 & 0.5000 \\0.5000 & {- 0.5000} & {0 + {0.5000\; i}}\end{matrix}$ $C_{10,3} = \begin{matrix}0.5000 & {- 0.5000} & {- 0.5000} \\0.5000 & {0 - {0.5000\; i}} & {0 + {0.5000\; i}} \\0.5000 & 0.5000 & 0.5000 \\0.5000 & {0 + {0.5000\; i}} & {0 - {0.5000\; i}}\end{matrix}$ $C_{11,3} = \begin{matrix}0.5000 & 0.5000 & {- 0.5000} \\{0 + {0.5000\; i}} & {0 - {0.5000\; i}} & {- 0.5000} \\0.5000 & 0.5000 & 0.5000 \\{0 + {0.5000\; i}} & {0 - {0.5000\; i}} & 0.5000\end{matrix}$ $C_{12,3} = \begin{matrix}0.5000 & {- 0.5000} & {- 0.5000} \\{0 + {0.5000\; i}} & {- 0.5000} & 0.5000 \\0.5000 & 0.5000 & 0.5000 \\{0 + {0.5000\; i}} & 0.5000 & {- 0.5000}\end{matrix}$ $C_{13,3} = \begin{matrix}0.5000 & 0.5000 & 0.5000 \\0.5000 & {0 + {0.5000\; i}} & {- 0.5000} \\0.5000 & {- 0.5000} & 0.5000 \\{- 0.5000} & {0 + {0.5000\; i}} & 0.5000\end{matrix}$ $C_{14,3} = \begin{matrix}0.5000 & 0.5000 & 0.5000 \\0.5000 & {- 0.5000} & {0 - {0.5000\; i}} \\0.5000 & 0.5000 & {- 0.5000} \\{- 0.5000} & 0.5000 & {0 - {0.5000\; i}}\end{matrix}$

$C_{15,3} = \begin{matrix}0.5000 & 0.5000 & 0.5000 \\{0.3536 + {0.3536\; i}} & {{- 0.3536} + {0.3536\; i}} & {0.3536 - {0.3536\; i}} \\{0 + {0.5000\; i}} & {0 - {0.5000\; i}} & {0 - {0.5000\; i}} \\{{- 0.3536} + {0.3536\; i}} & {0.3536 + {0.3536\; i}} & {{- 0.3536} - {0.3536\; i}}\end{matrix}$ $C_{16,3} = \begin{matrix}0.5000 & 0.5000 & 0.5000 \\{{- 0.3536} + {0.3536\; i}} & {{- 0.3536} - {0.3536\; i}} & {0.3536 - {0.3536\; i}} \\{0 - {0.5000\; i}} & {0 + {0.5000\; i}} & {0 - {0.5000\; i}} \\{0.3536 + {0.3536\; i}} & {0.3536 - {0.3536\; i}} & {{- 0.3536} - {0.3536\; i}}\end{matrix}$ $C_{1,4} = \begin{matrix}0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\0.5000 & {- 0.5000} & {- 0.5000} & 0.5000 \\0.5000 & 0.5000 & 0.5000 & 0.5000 \\0.5000 & {- 0.5000} & 0.5000 & {- 0.5000}\end{matrix}$ $C_{2,4} = \begin{matrix}0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\{0 + {0.5000\; i}} & {0 - {0.5000\; i}} & {0 - {0.5000\; i}} & {0 + {0.5000\; i}} \\0.5000 & 0.5000 & 0.5000 & 0.5000 \\{0 + {0.5000\; i}} & {0 - {0.5000\; i}} & {0 + {0.5000\; i}} & {0 - {0.5000\; i}}\end{matrix}$ $C_{3,4} = \begin{matrix}0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\0.5000 & {- 0.5000} & {0 - {0.5000\; i}} & {0 + {0.5000\; i}} \\0.5000 & 0.5000 & 0.5000 & 0.5000 \\0.5000 & {- 0.5000} & {0 + {0.5000\; i}} & {0 - {0.5000\; i}}\end{matrix}$ $C_{4,4} = \begin{matrix}0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\{0 + {0.5000\; i}} & {0 - {0.5000\; i}} & {- 0.5000} & 0.5000 \\0.5000 & 0.5000 & 0.5000 & 0.5000 \\{0 + {0.5000\; i}} & {0 - {0.5000\; i}} & 0.5000 & {- 0.5000}\end{matrix}$

$C_{5,4} = \begin{matrix}0.5000 & 0.5000 & 0.5000 & 0.5000 \\0.5000 & {0 + {0.5000\; i}} & {- 0.5000} & {0 - {0.5000\; i}} \\0.5000 & {- 0.5000} & 0.5000 & {- 0.5000} \\{- 0.5000} & {0 + {0.5000\; i}} & 0.5000 & {0 - {0.5000\; i}}\end{matrix}$ $C_{6,4} = \begin{matrix}0.5000 & 0.5000 & 0.5000 & 0.5000 \\{0.3536 + {0.3536\; i}} & {{- 0.3536} + {0.3536\; i}} & {{- 0.3536} - {0.3536\; i}} & {0.3536 + {0.3536\; i}} \\{0 + {0.5000\; i}} & {0 - {0.5000\; i}} & {0 + {0.5000\; i}} & {0 - {0.5000\; i}} \\{{- 0.3536} + {0.3536\; i}} & {0.3536 + {0.3536\; i}} & {0.3536 - {0.3536\; i}} & {{- 0.3536} - {0.3536\; i}}\end{matrix}$

3-2) Codebook used in a downlink of a multi-user MIMO communicationsystem performing unitary precoding where a number of physical antennasof a base station is four:

For example, where the number of physical antennas of the base stationis four, the codebook used in the downlink of the multi-user MIMOcommunication system according to an exemplary embodiment may bedesigned using the following Equation 8:

$\begin{matrix}{{W_{3} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix}1 & 1 & 0 & 0 \\1 & {- 1} & 0 & 0 \\0 & 0 & 1 & 1 \\0 & 0 & j & {- j}\end{bmatrix}}}{and}\begin{matrix}{W_{6} = {{{diag}( {1,\frac{( {1 + j} )}{\sqrt{2}},j,\frac{( {{- 1} + j} )}{\sqrt{2}}} )}*D\; F\; T}} \\{= {0.5*{\begin{bmatrix}1 & 1 & 1 & 1 \\\frac{( {1 + j} )}{\sqrt{2}} & \frac{( {{- 1} + j} )}{\sqrt{2}} & \frac{( {{- 1} - j} )}{\sqrt{2}} & \frac{( {1 - j} )}{\sqrt{2}} \\j & {- j} & j & {- j} \\\frac{( {{- 1} + j} )}{\sqrt{2}} & \frac{( {1 + j} )}{\sqrt{2}} & \frac{( {1 - j} )}{\sqrt{2}} & \frac{( {{- 1} - j} )}{\sqrt{2}}\end{bmatrix}.}}}\end{matrix}} & (8)\end{matrix}$

The codewords included in the codebook for the multi-user MIMOcommunication system performing unitary precoding, may be given by thefollowing Table 4:

codeword used at Transmit Codebook terminal for Index quantization 1 M₁= W3(;, 1) 2 M₂ = W3(;, 2) 3 M₃ = W3(;, 3) 4 M₄ = W3(;, 4) 5 M₅ =W6(;, 1) 6 M₆ = W6(;, 2) 7 M₇ = W6(;, 3) 8 M₈ = W6(;, 4)

Referring to the above Table 4, where the transmission rank of each ofusers is 1, the precoding matrix may be constructed by appropriatelycombining W₃(;,1), W₃(;,2), W₃(;,3), W₃(;,4), W₆(;,1), W₆(;,2), W₆(;,3),and W₆(;,4). Here, W_(k)(;,n) denotes the n^(th) column vector of W_(k).

The codewords included in the above Table 4 may be expressed as follows:

$M_{1} = \begin{matrix}0.5000 \\0.5000 \\0.5000 \\0.5000\end{matrix}$ $M_{2} = \begin{matrix}0.5000 \\{- 0.5000} \\0.5000 \\{- 0.5000}\end{matrix}$ $M_{3} = \begin{matrix}{- 0.5000} \\{0 - {0.5000\; i}} \\0.5000 \\{0 + {0.5000\; i}}\end{matrix}$ $M_{4} = \begin{matrix}{- 0.5000} \\{0 + {0.5000\; i}} \\0.5000 \\{0 - {0.5000\; i}}\end{matrix}$

$M_{5} = \begin{matrix}0.5000 \\{0.3536 + {0.3536\; i}} \\{0 + {0.5000\; i}} \\{{- 0.3536} + {0.3536\; i}}\end{matrix}$ $M_{6} = \begin{matrix}0.5000 \\{{- 0.3536} + {0.3536\; i}} \\{0 - {0.5000\; i}} \\{0.3536 + {0.3536\; i}}\end{matrix}$ $M_{7} = \begin{matrix}0.5000 \\{{- 0.3536} - {0.3536\; i}} \\{0 + {0.5000\; i}} \\{0.3536 - {0.3536\; i}}\end{matrix}$ $M_{8} = \begin{matrix}0.5000 \\{0.3536 - {0.3536\; i}} \\{0 - {0.5000\; i}} \\{{- 0.3536} - {0.3536\; i}}\end{matrix}$

3-3) Codebook used in a downlink of a multi-user MIMO communicationsystem performing non-unitary precoding where a number of physicalantennas of a base station is four:

For example, codewords included in the codebook used in the downlink ofthe multi-user MIMO communication system performing non-unitaryprecoding may be given by the following Table 5. Here, in the multi-userMIMO communication system, a rank of each of the users is 1:

Transmit Codebook Index Rank 1 1 C_(1,1) = W1(;, 2) 2 C_(2,1) = W1(;, 3)3 C_(3,1) = W1(;, 4) 4 C_(4,1) = W2(;, 2) 5 C_(5,1) = W2(;, 3) 6 C_(6,1)= W2(;, 4) 7 C_(7,1) = W3(;, 1) 8 C_(8,1) = W4(;, 1) 9 C_(9,1) =W5(;, 1) 10 C_(10,1) = W5(;, 2) 11 C_(11,1) = W5(;, 3) 12 C_(12,1) =W5(;, 4) 13 C_(13,1) = W6(;, 1) 14 C_(14,1) = W6(;, 2) 15 C_(15,1) =W6(;, 3) 16 C_(16,1) = W6(;, 4)

The codewords included in the above Table 5 may be expressed as follows:

$C_{1,1} = \begin{matrix}0.5000 \\{- 0.5000} \\0.5000 \\{- 0.5000}\end{matrix}$ $C_{2,1} = \begin{matrix}{- 0.5000} \\{- 0.5000} \\0.5000 \\0.5000\end{matrix}$ $C_{3,1} = \begin{matrix}{- 0.5000} \\0.5000 \\0.5000 \\{- 0.5000}\end{matrix}$

$C_{4,1} = \begin{matrix}0.5000 \\{0 - {0.5000\; i}} \\0.5000 \\{0 - {0.5000\; i}}\end{matrix}$ $C_{5,1} = \begin{matrix}{- 0.5000} \\{0 - {0.5000\; i}} \\0.5000 \\{0 + {0.5000\; i}}\end{matrix}$ $C_{6,1} = \begin{matrix}{- 0.5000} \\{0 + {0.5000\; i}} \\0.5000 \\{0 - {0.5000\; i}}\end{matrix}$ $C_{7,1} = \begin{matrix}0.5000 \\0.5000 \\0.5000 \\0.5000\end{matrix}$ $C_{8,1} = \begin{matrix}0.5000 \\{0 + {0.5000\; i}} \\0.5000 \\{0 + {0.5000\; i}}\end{matrix}$ $C_{9,1} = \begin{matrix}0.5000 \\0.5000 \\0.5000 \\{- 0.5000}\end{matrix}$

$C_{10,1} = \begin{matrix}0.5000 \\{0 + {0.5000\; i}} \\{- 0.5000} \\{0 + {0.5000\; i}}\end{matrix}$ $C_{11,1} = \begin{matrix}0.5000 \\{- 0.5000} \\0.5000 \\0.5000\end{matrix}$ $C_{12,1} = \begin{matrix}0.5000 \\{0 - {0.5000\; i}} \\{- 0.5000} \\{0 - {0.5000\; i}}\end{matrix}$ $C_{,13,1} = \begin{matrix}0.5000 \\{0.3536 + {0.3536\; i}} \\{0 + {0.5000\; i}} \\{{- 0.3536} + {0.3536\; i}}\end{matrix}$ $C_{14,1} = \begin{matrix}0.5000 \\{{- 0.3536} + {0.3536\; i}} \\{0 - {0.5000\; i}} \\{0.3536 + {0.3536\; i}}\end{matrix}$ $C_{15,1} = \begin{matrix}0.5000 \\{{- 0.3536} - {0.3536\; i}} \\{0 + {0.5000\; i}} \\{0.3536 - {0.3536\; i}}\end{matrix}$ $C_{16,1} = \begin{matrix}0.5000 \\{0.3536 - {0.3536\; i}} \\{0 - {0.5000\; i}} \\{{- 0.3536} - {0.3536\; i}}\end{matrix}$

4) A second example of a codebook used in a downlink of a single userMIMO communication system or a multi-user MIMO communication systemwhere a number of physical antennas of a base station is four:

4-1) Codebook used in a downlink of a single user MIMO communicationsystem where a number of physical antennas of a base station is four:

Here, a rotation matrix Urot and a QPSK DFT matrix may be defined asfollows.

${U_{rot} = {\frac{1}{\sqrt{2}}\begin{bmatrix}1 & 0 & {- 1} & 0 \\0 & 1 & 0 & {- 1} \\1 & 0 & 1 & 0 \\0 & 1 & 0 & 1\end{bmatrix}}},{and}$ ${{D\; F\; T} = {0.5*\begin{bmatrix}1 & 1 & 1 & 1 \\1 & j & {- 1} & {- j} \\1 & {- 1} & 1 & {- 1} \\1 & {- j} & {- 1} & j\end{bmatrix}}},$diag(a, b, c, d) is a 4×4 matrix, and diagonal elements of diag(a, b, c,d) are a, b, c, and d, and all the remaining elements are zero.

Codewords W₁, W₂, W₃, W₄, W₅, and W₆ may be defined as follows:

$\begin{matrix}{W_{1} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix}1 & 1 & 0 & 0 \\1 & {- 1} & 0 & 0 \\0 & 0 & 1 & 1 \\0 & 0 & 1 & {- 1}\end{bmatrix}*\begin{bmatrix}1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & {- 1} & 0 \\0 & 0 & 0 & {- 1}\end{bmatrix}}} \\{= {0.5*\begin{bmatrix}1 & 1 & 1 & 1 \\1 & {- 1} & 1 & {- 1} \\1 & 1 & {- 1} & {- 1} \\1 & {- 1} & {- 1} & 1\end{bmatrix}}} \\{W_{2} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix}1 & 1 & 0 & 0 \\j & {- j} & 0 & 0 \\0 & 0 & 1 & 1 \\0 & 0 & j & {- j}\end{bmatrix}*\begin{bmatrix}1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & {- 1} & 0 \\0 & 0 & 0 & {- 1}\end{bmatrix}}} \\{= {0.5*\begin{bmatrix}1 & 1 & 1 & 1 \\j & {- j} & j & {- j} \\1 & 1 & {- 1} & {- 1} \\j & {- j} & {- j} & j\end{bmatrix}}} \\{W_{3} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix}1 & 1 & 0 & 0 \\1 & {- 1} & 0 & 0 \\0 & 0 & 1 & 1 \\0 & 0 & j & {- j}\end{bmatrix}*\begin{bmatrix}1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & {- 1} & 0 \\0 & 0 & 0 & {- 1}\end{bmatrix}}} \\{= {0.5*\begin{bmatrix}1 & 1 & 1 & 1 \\1 & {- 1} & j & {- j} \\1 & 1 & {- 1} & {- 1} \\1 & {- 1} & {- j} & j\end{bmatrix}}} \\{W_{4} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix}1 & 1 & 0 & 0 \\j & {- j} & 0 & 0 \\0 & 0 & 1 & 1 \\0 & 0 & 1 & {- 1}\end{bmatrix}*\begin{bmatrix}1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & {- 1} & 0 \\0 & 0 & 0 & {- 1}\end{bmatrix}}} \\{= {0.5*\begin{bmatrix}1 & 1 & 1 & 1 \\j & {- j} & 1 & {- 1} \\1 & 1 & {- 1} & {- 1} \\j & {- j} & {- 1} & 1\end{bmatrix}}} \\{W_{5} = {{{diag}( {1,1,1,{- 1}} )}*D\; F\; T}} \\{= {0.5*\begin{bmatrix}1 & 1 & 1 & 1 \\1 & j & {- 1} & {- j} \\1 & {- 1} & 1 & {- 1} \\{- 1} & j & 1 & {- j}\end{bmatrix}}} \\{W_{6} = {{{diag}( {1,\frac{( {1 + j} )}{\sqrt{2}},j,\frac{( {{- 1} + j} )}{\sqrt{2}}} )}*D\; F\; T}} \\{= {0.5*\begin{bmatrix}1 & 1 & 1 & 1 \\\frac{( {1 + j} )}{\sqrt{2}} & \frac{( {{- 1} + j} )}{\sqrt{2}} & \frac{( {{- 1} - j} )}{\sqrt{2}} & \frac{( {1 - j} )}{\sqrt{2}} \\j & {- j} & j & {- j} \\\frac{( {{- 1} + j} )}{\sqrt{2}} & \frac{( {1 + j} )}{\sqrt{2}} & \frac{( {1 - j} )}{\sqrt{2}} & \frac{( {{- 1} - j} )}{\sqrt{2}}\end{bmatrix}}}\end{matrix}$

For example, where the number of physical antennas of the base stationis four, the matrices or the codewords included in the codebook used inthe downlink of the single user MIMO communication system according tothe second example may be given by the following Table 6:

Transmit Codebook Index Rank 1 Rank 2 Rank 3 Rank 4 1 C_(1,1) = W1(;, 2)C_(1,2) = W1(;, 2 1) C_(1,3) = W1(;, 1 2 3) C_(1,4) = W1(;, 1 2 3 4) 2C_(2,1) = W1(;, 3) C_(2,2) = W1(;, 3 1) C_(2,3) = W1(;, 1 2 4) C_(2,4) =W2(;, 1 2 3 4) 3 C_(3,1) = W1(;, 4) C_(3,2) = W1(;, 4 1) C_(3,3) = W1(;,1 3 4) C_(3,4) = W3(;, 1 2 3 4) 4 C_(4,1) = W2(;, 2) C_(4,2) = W1(;, 23) C_(4,3) = W1(;, 2 3 4) C_(4,4) = W4(;, 1 2 3 4) 5 C_(5,1) = W2(;, 3)C_(5,2) = W1(;, 2 4) C_(5,3) = W2(;, 1 2 3) C_(5,4) = W5(;, 1 2 3 4) 6C_(6,1) = W2(;, 4) C_(6,2) = W1(;, 3 4) C_(6,3) = W2(;, 1 2 4) C_(6,4) =W6(;, 1 2 3 4) 7 C_(7,1) = W3(;, 1) C_(7,2) = W2(;, 3 1) C_(7,3) = W2(;,1 3 4) n/a 8 C_(8,1) = W4(;, 1) C_(8,2) = W2(;, 4 1) C_(8,3) = W2(;, 2 34) n/a 9 C_(9,1) = W5(;, 1) C_(9,2) = W2(;, 2 3) C_(9,3) = W5(;, 1 2 3)n/a 10 C_(10,1) = W5(;, 2) C_(10,2) = W2(;, 2 4) C_(10,3) = W5(;, 1 2 4)n/a 11 C_(11,1) = W5(;, 3) C_(11,2) = W3(;, 3 1) C_(11,3) = W5(;, 1 3 4)n/a 12 C_(12,1) = W5(;, 4) C_(12,2) = W3(;, 4 1) C_(12,3) = W5(;, 2 3 4)n/a 13 C_(13,1) = W6(;, 1) C_(13,2) = W4(;, 3 1) C_(13,3) = W6(;, 1 2 3)n/a 14 C_(14,1) = W6(;, 2) C_(14,2) = W4(;, 4 1) C_(14,3) = W6(;, 1 2 4)n/a 15 C_(15,1) = W6(;, 3) C_(15,2) = W5(;, 1 3) C_(15,3) = W6(;, 1 3 4)n/a 16 C_(16,1) = W6(;, 4) C_(16,2) = W6(;, 2 4) C_(16,3) = W6(;, 2 3 4)n/a

The codewords included in the above Table 6 may be expressed as follows:

$C_{1,1} = \begin{matrix}0.5000 \\{- 0.5000} \\0.5000 \\{- 0.5000}\end{matrix}$ $C_{2,1} = \begin{matrix}0.5000 \\0.5000 \\{- 0.5000} \\{- 0.5000}\end{matrix}$ $C_{3,1} = \begin{matrix}0.5000 \\{- 0.5000} \\{- 0.5000} \\0.5000\end{matrix}$ $C_{4,1} = \begin{matrix}0.5000 \\{0 - {0.5000\; i}} \\0.5000 \\{0 - {0.5000\; i}}\end{matrix}$

$C_{5,1} = \begin{matrix}0.5000 \\{0 + {0.5000\; i}} \\{- 0.5000} \\{0 - {0.5000\; i}}\end{matrix}$ $C_{6,1} = \begin{matrix}0.5000 \\{0 - {0.5000\; i}} \\{- 0.5000} \\{0 + {0.5000\; i}}\end{matrix}$ $C_{7,1} = \begin{matrix}0.5000 \\0.5000 \\0.5000 \\0.5000\end{matrix}$ $C_{8,1} = \begin{matrix}0.5000 \\{0 + {0.5000\; i}} \\0.5000 \\{0 + {0.5000\; i}}\end{matrix}$ $C_{9,1} = \begin{matrix}0.5000 \\0.5000 \\0.5000 \\{- 0.5000}\end{matrix}$ $C_{10,1} = \begin{matrix}0.5000 \\{0 + {0.5000\; i}} \\{- 0.5000} \\{0 + {0.5000\; i}}\end{matrix}$

$C_{11,1} = \begin{matrix}0.5000 \\{- 0.5000} \\0.5000 \\0.5000\end{matrix}$ $C_{12,1} = \begin{matrix}0.5000 \\{0 - {0.5000\; i}} \\{- 0.5000} \\{0 - {0.5000\; i}}\end{matrix}$ $C_{13,1} = \begin{matrix}0.5000 \\{0.3536 + {0.3536\; i}} \\{0 + {0.5000\; i}} \\{{- 0.3536} + {0.3536\; i}}\end{matrix}$ $C_{14,1} = \begin{matrix}0.5000 \\{{- 0.3536} + {0.3536\; i}} \\{0 - {0.5000\; i}} \\{0.3536 + {0.3536\; i}}\end{matrix}$ $C_{,15,1} = \begin{matrix}0.5000 \\{{- 0.3536} + {0.3536\; i}} \\{0 + {0.5000\; i}} \\{0.3536 - {0.3536\; i}}\end{matrix}$ $C_{16,1} = \begin{matrix}0.5000 \\{0.3536 - {0.3536\; i}} \\{0 - {0.5000\; i}} \\{{- 0.3536} - {0.3536\; i}}\end{matrix}$

$C_{1.2} = \begin{matrix}0.5000 & 0.5000 \\{- 0.5000} & 0.5000 \\0.5000 & 0.5000 \\{- 0.5000} & 0.5000\end{matrix}$ $C_{2,2} = \begin{matrix}0.5000 & 0.5000 \\0.5000 & 0.5000 \\{- 0.5000} & 0.5000 \\{- 0.5000} & 0.5000\end{matrix}$ $C_{3,2} = \begin{matrix}0.5000 & 0.5000 \\{- 0.5000} & 0.5000 \\{- 0.5000} & 0.5000 \\0.5000 & 0.5000\end{matrix}$ $C_{4,2} = \begin{matrix}0.5000 & 0.5000 \\{- 0.5000} & 0.5000 \\0.5000 & {- 0.5000} \\{- 0.5000} & {- 0.5000}\end{matrix}$ $C_{5,2} = \begin{matrix}0.5000 & 0.5000 \\{- 0.5000} & {- 0.5000} \\0.5000 & {- 0.5000} \\{- 0.5000} & 0.5000\end{matrix}$ $C_{6,2} = \begin{matrix}0.5000 & 0.5000 \\0.5000 & {- 0.5000} \\{- 0.5000} & {- 0.5000} \\{- 0.5000} & 0.5000\end{matrix}$

$C_{7,2} = \begin{matrix}0.5000 & 0.5000 \\{0 + {0.5000\; i}} & {0 + {0.5000\; i}} \\{- 0.5000} & 0.5000 \\{0 - {0.5000\; i}} & {0 + {0.5000\; i}}\end{matrix}$ $C_{8,2} = \begin{matrix}0.5000 & 0.5000 \\{0 - {0.5000\; i}} & {0 + {0.5000\; i}} \\{- 0.5000} & 0.5000 \\{0 + {0.5000\; i}} & {0 + {0.5000\; i}}\end{matrix}$ $C_{9,2} = \begin{matrix}0.5000 & 0.5000 \\{0 - {0.5000\; i}} & {0 + {0.5000\; i}} \\0.5000 & {- 0.5000} \\{0 - {0.5000\; i}} & {0 - {0.5000\; i}}\end{matrix}$ $C_{10,2} = \begin{matrix}0.5000 & 0.5000 \\{0 - {0.5000\; i}} & {0 - {0.5000\; i}} \\0.5000 & {- 0.5000} \\{0 - {0.5000\; i}} & {0 + {0.5000\; i}}\end{matrix}$ $C_{11,2} = \begin{matrix}0.5000 & 0.5000 \\{0 + {0.5000\; i}} & 0.5000 \\{- 0.5000} & 0.5000 \\{0 - {0.5000\; i}} & 0.5000\end{matrix}$ $C_{12,2} = \begin{matrix}0.5000 & 0.5000 \\{0 - {0.5000\; i}} & 0.5000 \\{- 0.5000} & 0.5000 \\{0 + {0.5000\; i}} & 0.5000\end{matrix}$

$C_{13,2} = \begin{matrix}0.5000 & 0.5000 \\0.5000 & {0 + {0.5000\; i}} \\{- 0.5000} & 0.5000 \\{- 0.5000} & {0 + {0.5000\; i}}\end{matrix}$ $C_{14,2} = \begin{matrix}0.5000 & 0.5000 \\{- 0.5000} & {0 + {0.5000\; i}} \\{- 0.5000} & 0.5000 \\0.5000 & {0 + {0.5000\; i}}\end{matrix}$ $C_{15,2} = \begin{matrix}0.5000 & 0.5000 \\0.5000 & {- 0.5000} \\0.5000 & 0.5000 \\{- 0.5000} & 0.5000\end{matrix}$ $C_{16,2} = \begin{matrix}0.5000 & 0.5000 \\{{- 0.3536} + {0.3536\; i}} & {0.3536 - {0.3536\; i}} \\{0 - {0.5000\; i}} & {0 - {0.5000\; i}} \\{0.3536 + {0.3536\; i}} & {{- 0.3536} - {0.3536\; i}}\end{matrix}$ $C_{1,3} = \begin{matrix}0.5000 & 0.5000 & 0.5000 \\0.5000 & {- 0.5000} & 0.5000 \\0.5000 & 0.5000 & {- 0.5000} \\0.5000 & {- 0.5000} & {- 0.5000}\end{matrix}$ $C_{2,3} = \begin{matrix}0.5000 & 0.5000 & 0.5000 \\0.5000 & {- 0.5000} & {- 0.5000} \\0.5000 & 0.5000 & {- 0.5000} \\0.5000 & {- 0.5000} & 0.5000\end{matrix}$

$C_{3,3} = \begin{matrix}0.5000 & 0.5000 & 0.5000 \\0.5000 & 0.5000 & {- 0.5000} \\0.5000 & {- 0.5000} & {- 0.5000} \\0.5000 & {- 0.5000} & 0.5000\end{matrix}$ $C_{4,3} = \begin{matrix}0.5000 & 0.5000 & 0.5000 \\{- 0.5000} & 0.5000 & {- 0.5000} \\0.5000 & {- 0.5000} & {- 0.5000} \\{- 0.5000} & {- 0.5000} & 0.5000\end{matrix}$ $C_{5,3} = \begin{matrix}0.5000 & 0.5000 & 0.5000 \\{0 + {0.5000\; i}} & {0 - {0.5000\; i}} & {0 + {0.5000\; i}} \\0.5000 & 0.5000 & {- 0.5000} \\{0 + {0.5000\; i}} & {0 - {0.5000\; i}} & {0 - {0.5000\; i}}\end{matrix}$ $C_{6,3} = \begin{matrix}0.5000 & 0.5000 & 0.5000 \\{0 + {0.5000\; i}} & {0 - {0.5000\; i}} & {0 - {0.5000\; i}} \\0.5000 & 0.5000 & {- 0.5000} \\{0 + {0.5000\; i}} & {0 - {0.5000\; i}} & {0 + {0.5000\; i}}\end{matrix}$ $C_{7,3} = \begin{matrix}0.5000 & 0.5000 & 0.5000 \\{0 + {0.5000\; i}} & {0 + {0.5000\; i}} & {0 - {0.5000\; i}} \\0.5000 & {- 0.5000} & {- 0.5000} \\{0 + {0.5000\; i}} & {0 - {0.5000\; i}} & {0 + {0.5000\; i}}\end{matrix}$ $C_{8,3} = \begin{matrix}0.5000 & 0.5000 & 0.5000 \\{0 - {0.5000\; i}} & {0 + {0.5000\; i}} & {0 - {0.5000\; i}} \\0.5000 & {- 0.5000} & {- 0.5000} \\{0 - {0.5000\; i}} & {0 - {0.5000\; i}} & {0 + {0.5000\; i}}\end{matrix}$

$C_{9,3} = \begin{matrix}0.5000 & 0.5000 & 0.5000 \\0.5000 & {0 + {0.5000\; i}} & {- 0.5000} \\0.5000 & {- 0.5000} & 0.5000 \\{- 0.5000} & {0 + {0.5000\; i}} & 0.5000\end{matrix}$ $C_{10,3} = \begin{matrix}0.5000 & 0.5000 & 0.5000 \\0.5000 & {0 + {0.5000\; i}} & {0 - {0.5000\; i}} \\0.5000 & {- 0.5000} & {- 0.5000} \\{- 0.5000} & {0 + {0.5000\; i}} & {0 - {0.5000\; i}}\end{matrix}$ $C_{11,3} = \begin{matrix}0.5000 & 0.5000 & 0.5000 \\0.5000 & {- 0.5000} & {0 - {0.5000\; i}} \\0.5000 & 0.5000 & {- 0.5000} \\{- 0.5000} & 0.5000 & {0 - {0.5000\; i}}\end{matrix}$ $C_{12,3} = \begin{matrix}0.5000 & 0.5000 & 0.5000 \\{0 + {0.5000\; i}} & {- 0.5000} & {0 - {0.5000\; i}} \\{- 0.5000} & 0.5000 & {- 0.5000} \\{0 + {0.5000\; i}} & 0.5000 & {0 - {0.5000\; i}}\end{matrix}$ $C_{13,3} = \begin{matrix}0.5000 & 0.5000 & 0.5000 \\{0.3536 + {0.3536\; i}} & {{- 0.3536} + {0.3536\; i}} & {{- 0.3536} - {0.3536\; i}} \\{0 + {0.5000\; i}} & {0 - {0.5000\; i}} & {0 + {0.5000\; i}} \\{{- 0.3536} + {0.3536\; i}} & {0.3536 + {0.3536\; i}} & {0.3536 - {0.3536\; i}}\end{matrix}$ $C_{14,3} = \begin{matrix}0.5000 & 0.5000 & 0.5000 \\{0.3536 + {0.3536\; i}} & {{- 0.3536} + {0.3536\; i}} & {0.3536 - {0.3536\; i}} \\{0 + {0.5000\; i}} & {0 - {0.5000\; i}} & {0 - {0.5000\; i}} \\{{- 0.3536} + {0.3536\; i}} & {0.3536 + {0.3536\; i}} & {{- 0.3536} - {0.3536\; i}}\end{matrix}$

$C_{15,3} = \begin{matrix}0.5000 & 0.5000 & 0.5000 \\{0.3536 + {0.3536\; i}} & {{- 0.3536} - {0.3536\; i}} & {0.3536 - {0.3536\; i}} \\{0 + {0.5000\; i}} & {0 + {0.5000\; i}} & {0 - {0.5000\; i}} \\{{- 0.3536} + {0.3536\; i}} & {0.3536 - {0.3536\; i}} & {{- 0.3536} - {0.3536\; i}}\end{matrix}$ $C_{16,3} = \begin{matrix}0.5000 & 0.5000 & 0.5000 \\{{- 0.3536} + {0.3536\; i}} & {{- 0.3536} - {0.3536\; i}} & {0.3536 - {0.3536\; i}} \\{0 - {0.5000\; i}} & {0 + {0.5000\; i}} & {0 - {0.5000\; i}} \\{0.3536 + {0.3536\; i}} & {0.3536 - {0.3536\; i}} & {{- 0.3536} - {0.3536\; i}}\end{matrix}$ $C_{1,4} = \begin{matrix}0.5000 & 0.5000 & 0.5000 & 0.5000 \\0.5000 & {- 0.5000} & 0.5000 & {- 0.5000} \\0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\0.5000 & {- 0.5000} & {- 0.5000} & 0.5000\end{matrix}$ $C_{2,4} = \begin{matrix}0.5000 & 0.5000 & 0.5000 & 0.5000 \\{0 + {0.5000\; i}} & {0 - {0.5000\; i}} & {0 + {0.5000\; i}} & {0 - {0.5000\; i}} \\0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\{0 + {0.5000\; i}} & {0 - {0.5000\; i}} & {0 - {0.5000\; i}} & {0 + {0.5000\; i}}\end{matrix}$ $C_{3,4} = \begin{matrix}0.5000 & 0.5000 & 0.5000 & 0.5000 \\0.5000 & {- 0.5000} & {0 + {0.5000\; i}} & {0 - {0.5000\; i}} \\0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\0.5000 & {- 0.5000} & {0 - {0.5000\; i}} & {0 + {0.5000\; i}}\end{matrix}$ $C_{4,4} = \begin{matrix}0.5000 & 0.5000 & 0.5000 & 0.5000 \\{0 + {0.5000\; i}} & {0 - {0.5000\; i}} & 0.5000 & {- 0.5000} \\0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\{0 + {0.5000\; i}} & {0 - {0.5000\; i}} & {- 0.5000} & 0.5000\end{matrix}$

$C_{5,4} = \begin{matrix}0.5000 & 0.5000 & 0.5000 & 0.5000 \\0.5000 & {0 + {0.5000\; i}} & {- 0.5000} & {0 - {0.5000\; i}} \\0.5000 & {- 0.5000} & 0.5000 & {- 0.5000} \\{- 0.5000} & {0 + {0.5000\; i}} & 0.5000 & {0 - {0.5000\; i}}\end{matrix}$ $C_{6,4} = \begin{matrix}0.5000 & 0.5000 & 0.5000 & 0.5000 \\{0.3536 + {0.3536\; i}} & {{- 0.3536} + {0.3536\; i}} & {{- 0.3536} - {0.3536\; i}} & {0.3536 - {0.3536\; i}} \\{0 + {05000\; i}} & {0 - {0.5000\; i}} & {0 + {0.5000\; i}} & {0 - {05000\; i}} \\{{- 0.3536} + {0.3536\; i}} & {0.3536 + {0.3536\; i}} & {0.3536 - {0.3536\; i}} & {{- 0.3536} - {0.3536\; i}}\end{matrix}$

4-2) Codebook used in a downlink of a multi-user MIMO communicationsystem performing unitary precoding where a number of physical antennasof a base station is four:

For example, the codebook used in the downlink of the multi-user MIMOcommunication system according to the second example may be designed byappropriately combining two matrices W₃ and W₆ as follows:

$\begin{matrix}{W_{3} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix}1 & 1 & 0 & 0 \\1 & {- 1} & 0 & 0 \\0 & 0 & 1 & 1 \\0 & 0 & j & {- j}\end{bmatrix}}} \\{= {0.5*\begin{bmatrix}1 & 1 & {- 1} & {- 1} \\1 & {- 1} & {- j} & j \\1 & 1 & 1 & 1 \\1 & {- 1} & j & {- j}\end{bmatrix}}} \\{W_{6} = {{{diag}( {1,\frac{( {1 + j} )}{\sqrt{2}},j,\frac{( {{- 1} + j} )}{\sqrt{2}}} )}*D\; F\; T}} \\{= {0.5*\begin{bmatrix}1 & 1 & 1 & 1 \\\frac{( {1 + j} )}{\sqrt{2}} & \frac{( {{- 1} + j} )}{\sqrt{2}} & \frac{( {{- 1} - j} )}{\sqrt{2}} & \frac{( {1 - j} )}{\sqrt{2}} \\j & {- j} & j & {- j} \\\frac{( {{- 1} + j} )}{\sqrt{2}} & \frac{( {1 + j} )}{\sqrt{2}} & \frac{( {1 - j} )}{\sqrt{2}} & \frac{( {{- 1} - j} )}{\sqrt{2}}\end{bmatrix}}}\end{matrix}$

For example, where the number of physical antennas of the base stationis four, the matrices or the codewords included in the codebook used inthe downlink of the multi-user MIMO communication system according tothe second example may be given by the following Table 7:

Transmit Codebook Index Rank 1 1 M₁ = W3(;, 1) 2 M₂ = W3(;, 2) 3 M₃ =W3(;, 3) 4 M₄ = W3(;, 4) 5 M₅ = W6(;, 1) 6 M₆ = W6(;, 2) 7 M₇ = W6(;, 3)8 M₈ = W6(;, 4)

The codewords included in the above Table 7 may be expressed as follows:

$M_{1} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\0.5000\end{matrix} \\0.5000\end{matrix} \\0.5000\end{matrix}$ $M_{2} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\{- 0.5000}\end{matrix} \\0.5000\end{matrix} \\{- 0.5000}\end{matrix}$ $M_{3} = \begin{matrix}\begin{matrix}\begin{matrix}{- 0.5000} \\{0 - {0.5000\; i}}\end{matrix} \\0.5000\end{matrix} \\{0 + {0.5000\; i}}\end{matrix}$ $M_{4} = \begin{matrix}\begin{matrix}\begin{matrix}{- 0.5000} \\{0 + {0.5000\; i}}\end{matrix} \\0.5000\end{matrix} \\{0 - {0.5000\; i}}\end{matrix}$

$M_{5} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\{0.3536 + {0.3536\; i}}\end{matrix} \\{0 + {0.5000\; i}}\end{matrix} \\{{- 0.3536} + {0.3536\; i}}\end{matrix}$ $M_{6} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\{{- 0.3536} + {0.3536\; i}}\end{matrix} \\{0 - {0.5000\; i}}\end{matrix} \\{0.3536 + {0.3536\; i}}\end{matrix}$ $M_{7} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\{{- 0.3536} - {0.3536\; i}}\end{matrix} \\{0 + {0.5000\; i}}\end{matrix} \\{0.3536 - {0.3536\; i}}\end{matrix}$ $M_{8} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\{0.3536 - {0.3536\; i}}\end{matrix} \\{0 - {0.5000\; i}}\end{matrix} \\{{- 0.3536} - {0.3536\; i}}\end{matrix}$

4-3) Codebook used in a downlink of a multi-user MIMO communicationsystem performing non-unitary precoding where a number of physicalantennas of a base station is four:

For example, where the number of physical antennas of the base stationis four, the codewords included in the codebook used in the downlink ofthe multi-user MIMO communication system performing non-unitaryprecoding may be given by the following Table 8:

Transmit Codebook Transmission Index Rank 1 1 C_(1,1) = W1(;, 2) 2C_(2,1) = W1(;, 3) 3 C_(3,1) = W1(;, 4) 4 C_(4,1) = W2(;, 2) 5 C_(5,1) =W2(;, 3) 6 C_(6,1) = W2(;, 4) 7 C_(7,1) = W3(;, 1) 8 C_(8,1) = W4(;, 1)9 C_(9,1) = W5(;, 1) 10 C_(10,1) = W5(;, 2) 11 C_(11,1) = W5(;, 3) 12C_(12,1) = W5(;, 4) 13 C_(13,1) = W6(;, 1) 14 C_(14,1) = W6(;, 2) 15C_(15,1) = W6(;, 3) 16 C_(16,1) = W6(;, 4)

The codewords included in the above Table 8 may be expressed as follows:

$C_{1,1} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\{- 0.5000}\end{matrix} \\0.5000\end{matrix} \\{- 0.5000}\end{matrix}$ $C_{2,1} = \begin{matrix}\begin{matrix}\begin{matrix}{- 0.5000} \\{- 0.5000}\end{matrix} \\0.5000\end{matrix} \\0.5000\end{matrix}$ $C_{3,1} = \begin{matrix}\begin{matrix}\begin{matrix}{- 0.5000} \\0.5000\end{matrix} \\0.5000\end{matrix} \\{- 0.5000}\end{matrix}$ $C_{4,1} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\{0 - {0.5000\; i}}\end{matrix} \\0.5000\end{matrix} \\{0 - {0.5000\; i}}\end{matrix}$

$C_{5,1} = \begin{matrix}\begin{matrix}\begin{matrix}{- 0.5000} \\{0 - {0.5000\; i}}\end{matrix} \\0.5000\end{matrix} \\{0 + {0.5000\; i}}\end{matrix}$ $C_{6,1} = \begin{matrix}\begin{matrix}\begin{matrix}{- 0.5000} \\{0 + {0.5000\; i}}\end{matrix} \\0.5000\end{matrix} \\{0 - {0.5000\; i}}\end{matrix}$ $C_{7,1} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\0.5000\end{matrix} \\0.5000\end{matrix} \\0.5000\end{matrix}$ $C_{8,1} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\{0 + {0.5000\; i}}\end{matrix} \\0.5000\end{matrix} \\{0 + {0.5000\; i}}\end{matrix}$ $C_{9,1} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\0.5000\end{matrix} \\0.5000\end{matrix} \\{- 0.5000}\end{matrix}$ $C_{10,1} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\{0 + {0.5000\; i}}\end{matrix} \\{- 0.5000}\end{matrix} \\{0 + {0.5000\; i}}\end{matrix}$

$C_{11,1} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\{- 0.5000}\end{matrix} \\0.5000\end{matrix} \\0.5000\end{matrix}$ $C_{12,1} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\{0 - {0.5000\; i}}\end{matrix} \\{- 0.5000}\end{matrix} \\{0 - {0.5000\; i}}\end{matrix}$ $C_{13,1} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\{0.3536 + {0.3536\; i}}\end{matrix} \\{0 + {0.5000\; i}}\end{matrix} \\{{- 0.3536} + {0.3536\; i}}\end{matrix}$ $C_{14,1} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\{{- 0.3536} + {0.3536\; i}}\end{matrix} \\{0 - {0.5000\; i}}\end{matrix} \\{0.3536 + {0.3536\; i}}\end{matrix}$ $C_{15,1} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\{{- 0.3536} - {0.3536\; i}}\end{matrix} \\{0 + {0.5000\; i}}\end{matrix} \\{0.3536 - {0.3536\; i}}\end{matrix}$ $C_{16,1} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\{0.3536 - {0.3536\; i}}\end{matrix} \\{0 - {0.5000\; i}}\end{matrix} \\{{- 0.3536} - {0.3536\; i}}\end{matrix}$

5) A third example of a codebook used in a downlink of a single userMIMO communication system or a multi-user MIMO communication systemwhere a number of physical antennas of a base station is four:

5-1) Codebook used in a downlink of a single user MIMO communicationsystem where a number of physical antennas of a base station is four:

Here, a rotation matrix Urot and a QPSK DFT matrix may be defined asfollows.

${U_{rot} = {\frac{1}{\sqrt{2}}\begin{bmatrix}1 & 0 & {- 1} & 0 \\0 & 1 & 0 & {- 1} \\1 & 0 & 1 & 0 \\0 & 1 & 0 & 1\end{bmatrix}}},{and}$ ${{D\; F\; T} = {0.5*\begin{bmatrix}1 & 1 & 1 & 1 \\1 & j & {- 1} & {- j} \\1 & {- 1} & 1 & {- 1} \\1 & {- j} & {- 1} & j\end{bmatrix}}},$diag(a, b, c, d) is a 4×4 matrix, and diagonal elements of diag(a, b, c,d) are a, b, c, and d, and all the remaining elements are zero.

Codewords W₁, W₂, W₃, W₄, W₅, and W₆ may be defined as follows:

$\begin{matrix}{W_{1} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix}1 & 1 & 0 & 0 \\1 & {- 1} & 0 & 0 \\0 & 0 & 1 & 1 \\0 & 0 & 1 & {- 1}\end{bmatrix}}} \\{= {0.5*\begin{bmatrix}1 & 1 & {- 1} & {- 1} \\1 & {- 1} & {- 1} & 1 \\1 & 1 & 1 & 1 \\1 & {- 1} & 1 & {- 1}\end{bmatrix}}} \\{W_{2} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix}1 & 1 & 0 & 0 \\j & {- j} & 0 & 0 \\0 & 0 & 1 & 1 \\0 & 0 & j & {- j}\end{bmatrix}}} \\{= {0.5*\begin{bmatrix}1 & 1 & {- 1} & {- 1} \\j & {- j} & {- j} & j \\1 & 1 & 1 & 1 \\j & {- j} & j & {- j}\end{bmatrix}}} \\{W_{3} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix}1 & 1 & 0 & 0 \\1 & {- 1} & 0 & 0 \\0 & 0 & 1 & 1 \\0 & 0 & j & {- j}\end{bmatrix}}} \\{= {0.5*\begin{bmatrix}1 & 1 & {- 1} & {- 1} \\1 & {- 1} & {- j} & j \\1 & 1 & 1 & 1 \\1 & {- 1} & j & {- j}\end{bmatrix}}} \\{W_{4} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix}1 & 1 & 0 & 0 \\j & {- j} & 0 & 0 \\0 & 0 & 1 & 1 \\0 & 0 & 1 & {- 1}\end{bmatrix}}} \\{= {0.5*\begin{bmatrix}1 & 1 & {- 1} & {- 1} \\j & {- j} & {- 1} & 1 \\1 & 1 & 1 & 1 \\j & {- j} & 1 & {- 1}\end{bmatrix}}} \\{W_{5} = {{{diag}( {1,1,1,{- 1}} )}*D\; F\; T}} \\{= {0.5*\begin{bmatrix}1 & 1 & 1 & 1 \\1 & j & {- 1} & {- j} \\1 & {- 1} & 1 & {- 1} \\{- 1} & j & 1 & {- j}\end{bmatrix}}} \\{W_{6} = {{{diag}( {1,\frac{( {1 + j} )}{\sqrt{2}},j,\frac{( {{- 1} + j} )}{\sqrt{2}}} )}*D\; F\; T}} \\{= {0.5*\begin{bmatrix}1 & 1 & 1 & 1 \\\frac{( {1 + j} )}{\sqrt{2}} & \frac{( {{- 1} + j} )}{\sqrt{2}} & \frac{( {{- 1} - j} )}{\sqrt{2}} & \frac{( {1 - j} )}{\sqrt{2}} \\j & {- j} & j & {- j} \\\frac{( {{- 1} + j} )}{\sqrt{2}} & \frac{( {1 + j} )}{\sqrt{2}} & \frac{( {1 - j} )}{\sqrt{2}} & \frac{( {{- 1} - j} )}{\sqrt{2}}\end{bmatrix}}}\end{matrix}$

For example, where the number of physical antennas of the base stationis four, the matrices or the codewords included in the codebook used inthe downlink of the single user MIMO communication system according tothe third example may be given by the following Table 9:

Transmit Codebook Index Rank 1 Rank 2 Rank 3 Rank 4 1 C_(1,1) = W1(;, 2)C_(1,2) = W1(;, 2 1) C_(1,3) = W1(;, 1 2 3) C_(1,4) = W1(;, 1 2 3 4) 2C_(2,1) = W1(;, 3) C_(2,2) = W1(;, 3 1) C_(2,3) = W1(;, 1 2 4) C_(2,4) =W2(;, 1 2 3 4) 3 C_(3,1) = W1(;, 4) C_(3,2) = W1(;, 4 1) C_(3,3) = W1(;,1 3 4) C_(3,4) = W3(;, 1 2 3 4) 4 C_(4,1) = W2(;, 2) C_(4,2) = W1(;, 23) C_(4,3) = W1(;, 2 3 4) C_(4,4) = W4(;, 1 2 3 4) 5 C_(5,1) = W2(;, 3)C_(5,2) = W1(;, 2 4) C_(5,3) = W2(;, 1 2 3) C_(5,4) = W5(;, 1 2 3 4) 6C_(6,1) = W2(;, 4) C_(6,2) = W1(;, 3 4) C_(6,3) = W2(;, 1 2 4) C_(6,4) =W6(;, 1 2 3 4) 7 C_(7,1) = W3(;, 1) C_(7,2) = W2(;, 3 1) C_(7,3) = W2(;,1 3 4) n/a 8 C_(8,1) = W4(;, 1) C_(8,2) = W2(;, 4 1) C_(8,3) = W2(;, 2 34) n/a 9 C_(9,1) = W5(;, 1) C_(9,2) = W2(;, 2 3) C_(9,3) = W5(;, 1 2 3)n/a 10 C_(10,1) = W5(;, 2) C_(10,2) = W2(;, 2 4) C_(10,3) = W5(;, 1 2 4)n/a 11 C_(11,1) = W5(;, 3) C_(11,2) = W3(;, 3 1) C_(11,3) = W5(;, 1 3 4)n/a 12 C_(12,1) = W5(;, 4) C_(12,2) = W3(;, 4 1) C_(12,3) = W5(;, 2 3 4)n/a 13 C_(13,1) = W6(;, 1) C_(13,2) = W4(;, 3 1) C_(13,3) = W6(;, 1 2 3)n/a 14 C_(14,1) = W6(;, 2) C_(14,2) = W4(;, 4 1) C_(14,3) = W6(;, 1 2 4)n/a 15 C_(15,1) = W6(;, 3) C_(15,2) = W5(;, 1 3) C_(15,3) = W6(;, 1 3 4)n/a 16 C_(16,1) = W6(;, 4) C_(16,2) = W6(;, 2 4) C_(16,3) = W6(;, 2 3 4)n/a

The codewords included in the above Table 9 may be expressed as follows:

$C_{1,1} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\{- 0.5000}\end{matrix} \\0.5000\end{matrix} \\{- 0.5000}\end{matrix}$ $C_{2,1} = \begin{matrix}\begin{matrix}\begin{matrix}{- 0.5000} \\{- 0.5000}\end{matrix} \\0.5000\end{matrix} \\0.5000\end{matrix}$ $C_{3,1} = \begin{matrix}\begin{matrix}\begin{matrix}{- 0.5000} \\0.5000\end{matrix} \\0.5000\end{matrix} \\{- 0.5000}\end{matrix}$ $C_{4,1} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\{0 - {05000\; i}}\end{matrix} \\0.5000\end{matrix} \\{0 - {0.5000\; i}}\end{matrix}$ $C_{5,1} = \begin{matrix}\begin{matrix}\begin{matrix}{- 0.5000} \\{0 - {0.5000\; i}}\end{matrix} \\0.5000\end{matrix} \\{0 + {0.5000\; i}}\end{matrix}$

$C_{6,1} = \begin{matrix}\begin{matrix}\begin{matrix}{- 0.5000} \\{0 + {0.5000\; i}}\end{matrix} \\0.5000\end{matrix} \\{0 - {0.5000\; i}}\end{matrix}$ $C_{7,1} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\0.5000\end{matrix} \\0.5000\end{matrix} \\0.5000\end{matrix}$ $C_{8,1} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\{0 + {0.5000\; i}}\end{matrix} \\05000\end{matrix} \\{0 + {0.5000\; i}}\end{matrix}$ $C_{9,1} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\0.5000\end{matrix} \\0.5000\end{matrix} \\{- 0.5000}\end{matrix}$ $C_{10,1} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\{0 + {0.5000\; i}}\end{matrix} \\{- 0.5000}\end{matrix} \\{0 + {0.5000\; i}}\end{matrix}$ $C_{11,1} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\{- 0.5000}\end{matrix} \\0.5000\end{matrix} \\0.5000\end{matrix}$

$C_{12,1} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\{0 - {0.5000\; i}}\end{matrix} \\{- 0.5000}\end{matrix} \\{0 - {0.5000\; i}}\end{matrix}$ $C_{13,1} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\{0.3536 + {0.3536\; i}}\end{matrix} \\{0 + {0.5000\; i}}\end{matrix} \\{{- 0.3536} + {0.3536\; i}}\end{matrix}$ $C_{14,1} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\{{- 0.3536} + {0.3536\; i}}\end{matrix} \\{0 - {0.5000\; i}}\end{matrix} \\{0.3536 + {0.3536\; i}}\end{matrix}$ $C_{15,1} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\{{- 0.3536} - {0.3536\; i}}\end{matrix} \\{0 + {0.5000\; i}}\end{matrix} \\{0.3536 - {0.3536\; i}}\end{matrix}$ $C_{16,1} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\{0.3536 - {0.3536\; i}}\end{matrix} \\{0 - {0.5000\; i}}\end{matrix} \\{{- 0.3536} - {0.3536\; i}}\end{matrix}$ $C_{1,2} = \begin{matrix}0.5000 & 0.5000 \\0.5000 & {- 0.5000} \\0.5000 & 0.5000 \\0.5000 & {- 0.5000}\end{matrix}$

$C_{2,2} = \begin{matrix}0.5000 & {- 0.5000} \\0.5000 & {- 0.5000} \\0.5000 & 0.5000 \\0.5000 & 0.5000\end{matrix}$ $C_{3,2} = \begin{matrix}0.5000 & {- 0.5000} \\0.5000 & 0.5000 \\0.5000 & 0.5000 \\0.5000 & {- 0.5000}\end{matrix}$ $C_{4,2} = \begin{matrix}0.5000 & {- 0.5000} \\{- 0.5000} & {- 0.5000} \\0.5000 & 0.5000 \\{- 0.5000} & 0.5000\end{matrix}$ $C_{5,2} = \begin{matrix}0.5000 & {- 0.5000} \\{- 0.5000} & 0.5000 \\0.5000 & 0.5000 \\{- 0.5000} & {- 0.5000}\end{matrix}$ $C_{6,2} = \begin{matrix}{- 0.5000} & {- 0.5000} \\{- 0.5000} & 0.5000 \\0.5000 & 0.5000 \\0.5000 & {- 0.5000}\end{matrix}$ $C_{7,2} = \begin{matrix}0.5000 & {- 0.5000} \\{0 + {0.5000\; i}} & {0 - {0.5000\; i}} \\0.5000 & 0.5000 \\{0 + {0.5000\; i}} & {0 + {0.5000\; i}}\end{matrix}$

$C_{8,2} = \begin{matrix}0.5000 & {- 0.5000} \\{0 + {0.5000\; i}} & {0 + {0.5000\; i}} \\0.5000 & 0.5000 \\{0 + {0.5000\; i}} & {0 - {0.5000\; i}}\end{matrix}$ $C_{9,2} = \begin{matrix}0.5000 & {- 0.5000} \\{0 - {0.5000\; i}} & {{0 - {0.5000\; i}}\;} \\0.5000 & 0.5000 \\{0 - {0.5000\; i}} & {0 + {0.5000\; i}}\end{matrix}$ $C_{10,2} = \begin{matrix}0.5000 & {- 0.5000} \\{0 - {0.5000\; i}} & {0 + {0.5000\; i}} \\0.5000 & 0.5000 \\{0 - {0.5000\; i}} & {0 - {0.5000\; i}}\end{matrix}$ $C_{11,2} = \begin{matrix}0.5000 & {- 0.5000} \\0.5000 & {0 - {0.5000\; i}} \\0.5000 & 0.5000 \\0.5000 & {0 + {0.5000\; i}}\end{matrix}$ $C_{12,2} = \begin{matrix}0.5000 & {- 0.5000} \\0.5000 & {0 + {0.5000\; i}} \\0.5000 & 0.5000 \\0.5000 & {0 - {0.5000\; i}}\end{matrix}$ $C_{13,2} = \begin{matrix}0.5000 & {- 0.5000} \\{0 + {0.5000\; i}} & {- 0.5000} \\0.5000 & 0.5000 \\{0 + {0.5000\; i}} & 0.5000\end{matrix}$

$C_{14,2} = \begin{matrix}0.5000 & {- 0.5000} \\{0 + {0.5000\; i}} & 0.5000 \\0.5000 & 0.5000 \\{0 + {0.5000\; i}} & {- 0.5000}\end{matrix}$ $C_{15,2} = \begin{matrix}0.5000 & 0.5000 \\0.5000 & {- 0.5000} \\0.5000 & 0.5000 \\{- 0.5000} & 0.5000\end{matrix}$ $C_{16,2} = \begin{matrix}0.5000 & 0.5000 \\{{- 0.3536} + {0.3536\; i}} & {0.3536 - {0.3536\; i}} \\{0 - {0.5000\; i}} & {0 - {0.5000\; i}} \\{0.3536 + {0.3536\; i}} & {{- 0.3536} - {0.3536\; i}}\end{matrix}$ $C_{1,3} = \begin{matrix}0.5000 & 0.5000 & {- 0.5000} \\0.5000 & {- 0.5000} & {- 0.5000} \\0.5000 & 0.5000 & 0.5000 \\0.5000 & {- 0.5000} & 0.5000\end{matrix}$ $C_{2,3} = \begin{matrix}0.5000 & 0.5000 & {- 0.5000} \\0.5000 & {- 0.5000} & 0.5000 \\0.5000 & 0.5000 & 0.5000 \\0.5000 & {- 0.5000} & {- 0.5000}\end{matrix}$ $C_{3,3} = \begin{matrix}0.5000 & {- 0.5000} & {- 0.5000} \\0.5000 & {- 0.5000} & 0.5000 \\0.5000 & 0.5000 & 0.5000 \\0.5000 & 0.5000 & {- 0.5000}\end{matrix}$

$C_{4,3} = \begin{matrix}0.5000 & {- 0.5000} & {- 0.5000} \\{- 0.5000} & {- 0.5000} & 0.5000 \\0.5000 & 0.5000 & 0.5000 \\{- 0.5000} & 0.5000 & {- 0.5000}\end{matrix}$ $C_{5,3} = \begin{matrix}0.5000 & 0.5000 & {- 0.5000} \\{0 + {0.5000i}} & {0 - {0.5000i}} & {0 - {0.5000i}} \\0.5000 & 0.5000 & 0.5000 \\{0 + {0.5000i}} & {0 - {0.5000i}} & {0 + {0.5000i}}\end{matrix}$ $C_{6,3} = \begin{matrix}0.5000 & 0.5000 & {- 0.5000} \\{0 + {0.5000i}} & {0 - {0.5000i}} & {0 + {0.5000i}} \\0.5000 & 0.5000 & 0.5000 \\{0 + {0.5000i}} & {0 - {0.5000i}} & {0 - {0.5000i}}\end{matrix}$ $C_{7,3} = \begin{matrix}0.5000 & {- 0.5000} & {- 0.5000} \\{0 + {0.5000i}} & {0 - {0.5000i}} & {0 + {0.5000i}} \\0.5000 & 0.5000 & 0.5000 \\{0 + {0.5000i}} & {0 + {0.5000i}} & {0 - {0.5000i}}\end{matrix}$ $C_{8,3} = \begin{matrix}0.5000 & {- 0.5000} & {- 0.5000} \\{0 - {0.5000i}} & {0 - {0.5000i}} & {0 + {0.5000i}} \\0.5000 & 0.5000 & 0.5000 \\{0 - {0.5000i}} & {0 + {0.5000i}} & {0 - {0.5000i}}\end{matrix}$ $C_{9,3} = \begin{matrix}0.5000 & 0.5000 & 0.5000 \\0.5000 & {0 - {0.5000i}} & {- 0.5000} \\0.5000 & {- 0.5000} & 0.5000 \\{- 0.5000} & {0 + {0.5000i}} & 0.5000\end{matrix}$

$\mspace{79mu}{C_{10,3} = \begin{matrix}0.5000 & 0.5000 & 0.5000 \\0.5000 & {0 + {0.5000i}} & {0 - 0.5000} \\0.5000 & {- 0.5000} & {- 0.5000} \\{- 0.5000} & {0 + {0.5000i}} & {0 - {0.5000i}}\end{matrix}}$ $\mspace{79mu}{C_{11,3} = \begin{matrix}0.5000 & 0.5000 & 0.5000 \\0.5000 & {- 0.5000} & {0 - {0.5000i}} \\0.5000 & 0.5000 & {- 0.5000} \\{- 0.5000} & 0.5000 & {0 - {0.5000i}}\end{matrix}}$ $\mspace{79mu}{C_{12,3} = \begin{matrix}0.5000 & 0.5000 & 0.5000 \\{0 + {0.5000i}} & {- 0.5000} & {0 - {0.5000i}} \\{- 0.5000} & 0.5000 & {- 0.5000} \\{0 + {0.5000i}} & 0.5000 & {0 - {0.5000i}}\end{matrix}}$ $C_{13,3} = \begin{matrix}0.5000 & 0.5000 & 0.5000 \\{0.3536 + {0.3536i}} & {{- 0.3536} + {0.3536i}} & {{- 0.3536} - {0.3536i}} \\{0 + {0.5000i}} & {0 - {0.5000i}} & {0 + {0.5000i}} \\{{- 0.3536} + {0.3536i}} & {0.3536 + {0.3536i}} & {0.3536 - {0.3536i}}\end{matrix}$ $C_{14,3} = \begin{matrix}0.5000 & 0.5000 & 0.5000 \\{0.3536 + {0.3536i}} & {{- 0.3536} + {0.3536i}} & {0.3536 - {0.3536i}} \\{0 + {0.5000i}} & {0 - {0.5000i}} & {0 - {0.5000i}} \\{{- 0.3536} + {0.3536i}} & {0.3536 + {0.3536i}} & {{- 0.3536} + {0.3536i}}\end{matrix}$ $C_{15,3} = \begin{matrix}0.5000 & 0.5000 & 0.5000 \\{0.3536 + {0.3536i}} & {{- 0.3536} - {0.3536i}} & {0.3536 - {0.3536i}} \\{0 + {0.5000i}} & {0 + {0.5000i}} & {0 - {0.5000i}} \\{{- 0.3536} + {0.3536i}} & {0.3536 - {0.3536i}} & {{- 0.3536} - {0.3536i}}\end{matrix}$

$C_{16,3} = \begin{matrix}0.5000 & 0.5000 & 0.5000 \\{{- 0.3536} + {0.3536i}} & {{- 0.3536} - {0.3536i}} & {0.3536 - {0.3536i}} \\{0 - {0.5000i}} & {0 + {0.5000i}} & {0 - {0.5000i}} \\{{- 0.3536} + {0.3536i}} & {0.3536 - {0.3536i}} & {{- 0.3536} - {0.3536i}}\end{matrix}$ $\mspace{79mu}{C_{1,4} = \begin{matrix}0.5000 & 0.5000 & {- 0.5000} \\0.5000 & {- 0.5000} & 0.5000 \\0.5000 & 0.5000 & 0.5000 \\0.5000 & {- 0.5000} & {- 0.5000}\end{matrix}}$ $\mspace{79mu}{C_{2,4} = \begin{matrix}0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\{0 + {0.5000i}} & {0 - {0.5000i}} & {0 - {0.5000i}} & {0 + 0.5000} \\0.5000 & 0.5000 & 0.5000 & 0.5000 \\{0 + {0.5000i}} & {0 - {0.5000i}} & {0 + {0.5000i}} & {0 - {0.5000i}}\end{matrix}}$ $\mspace{79mu}{C_{3,4} = \begin{matrix}0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\0.5000 & {- 0.5000} & {0 - {0.5000i}} & {0 + {0.5000i}} \\0.5000 & 0.5000 & 0.5000 & 0.5000 \\0.5000 & {- 0.5000} & {0 + {0.5000i}} & {0 - {0.5000i}}\end{matrix}}$ $\mspace{79mu}{C_{4,4} = \begin{matrix}0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\{0 + {0.5000i}} & {0 - {0.5000i}} & {- 0.5000} & 0.5000 \\0.5000 & 0.5000 & 0.5000 & 0.5000 \\{0 + {0.5000i}} & {0 - {0.5000i}} & 0.5000 & {- 0.5000}\end{matrix}}$ $\mspace{79mu}{C_{5,4} = \begin{matrix}0.5000 & 0.5000 & 0.5000 & 0.5000 \\0.5000 & {0 + {0.5000i}} & {- 0.5000} & {0 - {0.5000i}} \\0.5000 & {- 0.5000} & 0.5000 & {- 0.5000} \\{- 0.5000} & {0 + {0.5000i}} & 0.5000 & {0 - {0.5000i}}\end{matrix}}$

$C_{6,4} = \begin{matrix}0.5000 & 0.5000 & 0.5000 & 0.5000 \\{0.3536 + {0.3536i}} & {{- 0.3536} + {0.3536i}} & {{- 0.3536} - {0.3536i}} & {0.3536 - {0.3536i}} \\{0 + {0.5000i}} & {0 - {0.5000i}} & {0 + {0.5000i}} & {0 - {0.5000i}} \\{{- 0.3536} + {0.3536i}} & {0.3536 + {0.3536i}} & {{- 0.3536} - {0.3536i}} & {{- 0.3536} - {0.3536i}}\end{matrix}$

5-2) Codebook used in a downlink of a multi-user MIMO communicationsystem performing unitary precoding where a number of physical antennasof a base station is four:

For example, the codebook used in the downlink of the multi-user MIMOcommunication system according to the third example may be designed byappropriately combining two matrices W₃ and W₆ as follows:

$W_{3} = {{\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix}1 & 1 & 0 & 0 \\1 & {- 1} & 0 & 0 \\0 & 0 & 1 & 1 \\0 & 0 & j & {- j}\end{bmatrix}} = {0.5*\begin{bmatrix}1 & 1 & {- 1} & {- 1} \\1 & {- 1} & {- j} & j \\1 & 1 & 1 & 1 \\1 & {- 1} & j & {- j}\end{bmatrix}}}$ $\begin{matrix}{W_{6} = {{{diag}( {1,\frac{( {1 + j} )}{\sqrt{2}},j,\frac{( {{- 1} + j} )}{\sqrt{2}}} )}*D\; F\; T}} \\{= {0.5*\begin{bmatrix}1 & 1 & 1 & 1 \\\frac{( {1 + j} )}{\sqrt{2}} & \frac{( {{- 1} + j} )}{\sqrt{2}} & \frac{( {{- 1} - j} )}{\sqrt{2}} & \frac{( {1 - j} )}{\sqrt{2}} \\j & {- j} & j & {- j} \\\frac{( {{- 1} + j} )}{\sqrt{2}} & \frac{( {1 + j} )}{\sqrt{2}} & \frac{( {1 - j} )}{\sqrt{2}} & \frac{( {{- 1} - j} )}{\sqrt{2}}\end{bmatrix}}}\end{matrix}$

For example, where the number of physical antennas of the base stationis four, the matrices or the codewords included in the codebook used inthe downlink of the multi-user MIMO communication system according tothe third example may be given by the following Table 10:

Transmit Codebook Index Rank 1 1 M₁ = W3(;, 1) 2 M₂ = W3(;, 2) 3 M₃ =W3(;, 3) 4 M₄ = W3(;, 4) 5 M₅ = W6(;, 1) 6 M₆ = W6(;, 2) 7 M₇ = W6(;, 3)8 M₈ = W6(;, 4)

The codewords included in the above Table may be expressed as follows:

$M_{1} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\0.5000\end{matrix} \\0.5000\end{matrix} \\0.5000\end{matrix}$ $M_{2} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\{- 0.5000}\end{matrix} \\0.5000\end{matrix} \\{- 0.5000}\end{matrix}$ $M_{3} = \begin{matrix}\begin{matrix}\begin{matrix}{- 0.5000} \\{0 - {0.5000i}}\end{matrix} \\0.5000\end{matrix} \\{0 + {0.5000i}}\end{matrix}$ $M_{4} = \begin{matrix}\begin{matrix}\begin{matrix}{- 0.5000} \\{0 + {0.5000i}}\end{matrix} \\0.5000\end{matrix} \\{0 - {0.5000i}}\end{matrix}$

$M_{5} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\{0.3536 + {0.3536i}}\end{matrix} \\{0 + {0.5000i}}\end{matrix} \\{{- 0.3536} + {0.3536i}}\end{matrix}$ $M_{6} = \begin{matrix}\begin{matrix}\begin{matrix}{- 0.5000} \\{{- 0.3536} + {0.3536i}}\end{matrix} \\{0 - {0.5000i}}\end{matrix} \\{0.3536 + {0.3536i}}\end{matrix}$ $M_{7} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\{{- 0.3536} + {0.3536i}}\end{matrix} \\{0 - {0.5000i}}\end{matrix} \\{0.3536 + {0.3536i}}\end{matrix}$ $M_{8} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\{0.3536 - {0.3536i}}\end{matrix} \\{0 - {0.5000i}}\end{matrix} \\{{- 0.3536} - {0.3536i}}\end{matrix}$

5-3) Codebook used in a downlink of a multi-user MIMO communicationsystem performing non-unitary precoding where a number of physicalantennas of a base station is four:

For example, where the number of physical antennas of the base stationis four, the codewords included in the codebook used in the downlink ofthe multi-user MIMO communication system performing non-unitaryprecoding may be given by the following Table 11:

Transmit Codebook Transmission Index Rank 1 1 C_(1,1) = W1(;, 2) 2C_(2,1) = W1(;, 3) 3 C_(3,1) = W1(;, 4) 4 C_(4,1) = W2(;, 2) 5 C_(5,1) =W2(;, 3) 6 C_(6,1) = W2(;, 4) 7 C_(7,1) = W3(;, 1) 8 C_(8,1) = W4(;, 1)9 C_(9,1) = W5(;, 1) 10 C_(10,1) = W5(;, 2) 11 C_(11,1) = W5(;, 3) 12C_(12,1) = W5(;, 4) 13 C_(13,1) = W6(;, 1) 14 C_(14,1) = W6(;, 2) 15C_(15,1) = W6(;, 3) 16 C_(16,1) = W6(;, 4)

The codewords included in the above Table 11 may be expressed asfollows:

$C_{1,1} = \begin{matrix}0.5000 \\{- 0.5000} \\0.5000 \\{- 0.5000}\end{matrix}$ $C_{2,1} = \begin{matrix}{- 0.5000} \\{- 0.5000} \\0.5000 \\0.5000\end{matrix}$ $C_{3,1} = \begin{matrix}{- 0.5000} \\0.5000 \\0.5000 \\{- 0.5000}\end{matrix}$ $C_{4,1} = \begin{matrix}0.5000 \\{0 - {0.5000i}} \\0.5000 \\{0 - {0.5000i}}\end{matrix}$

$C_{5,1} = \begin{matrix}{- 0.5000} \\{0 - {0.5000i}} \\0.5000 \\{0 + {0.5000i}}\end{matrix}$ $C_{6,1} = \begin{matrix}{- 0.5000} \\{0 + {0.5000i}} \\0.5000 \\{0 - {0.5000i}}\end{matrix}$ $C_{7,1} = \begin{matrix}0.5000 \\0.5000 \\0.5000 \\0.5000\end{matrix}$ $C_{8,1} = \begin{matrix}0.5000 \\{0 + {0.5000i}} \\0.5000 \\{0 + {0.5000i}}\end{matrix}$ $C_{1,1} = \begin{matrix}0.5000 \\0.5000 \\0.5000 \\{- 0.5000}\end{matrix}$ $C_{10,1} = \begin{matrix}0.5000 \\{0 + {0.5000i}} \\{- 0.5000} \\{0 + {0.5000i}}\end{matrix}$

$C_{11,1} = \begin{matrix}0.5000 \\{- 0.5000} \\0.5000 \\0.5000\end{matrix}$ $C_{12,1} = \begin{matrix}0.5000 \\{0 - {0.5000i}} \\{- 0.5000} \\{0 - {0.5000i}}\end{matrix}$ $C_{13,1} = \begin{matrix}0.5000 \\{0.3536 + {0.3536i}} \\{0 + {0.5000i}} \\{{- 0.3536} + {0.3536i}}\end{matrix}$ $C_{14,1} = \begin{matrix}0.5000 \\{{- 0.3536} + {0.3536i}} \\{0 - {0.5000i}} \\{0.3536 + {0.3536i}}\end{matrix}$ $C_{15,1} = \begin{matrix}0.5000 \\{{- 0.3536} - {0.3536i}} \\{0 + {0.5000i}} \\{0.3536 - {0.3536i}}\end{matrix}$ $C_{16,1} = \begin{matrix}0.5000 \\{{- 0.3536} - {0.3536i}} \\{0 - {0.5000i}} \\{{- 0.3536} - {0.3536i}}\end{matrix}$

6) A fourth example of a codebook used in a downlink of a single userMIMO communication system or a multi-user MIMO communication systemwhere a number of physical antennas of a base station is four:

6-1) Codebook used in a downlink of a single user MIMO communicationsystem where a number of physical antennas of a base station is four:

Here, a rotation matrix Urot and a QPSK DFT matrix may be defined asfollows.

$U_{rot} = {{\frac{1}{\sqrt{2}}\begin{bmatrix}1 & 0 & {- 1} & 0 \\0 & 1 & 0 & {- 1} \\1 & 0 & 1 & 0 \\0 & 1 & 0 & 1\end{bmatrix}}\mspace{14mu}{and}}$ ${{D\; F\; T} = {0.5*\begin{bmatrix}1 & 1 & 1 & 1 \\1 & j & {- 1} & {- j} \\1 & {- 1} & 1 & {- 1} \\1 & {- j} & {- 1} & j\end{bmatrix}}},$diag(a, b, c, d) is a 4×4 matrix, and diagonal elements of diag(a, b, c,d) are a, b, c, and d, and all the remaining elements are zero.

Codewords W₁, W₂, W₃, W₄, W₅, and W₆ may be defined as follows:

$\begin{matrix}{W_{1} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix}1 & 1 & 0 & 0 \\1 & {- 1} & 0 & 0 \\0 & 0 & 1 & 1 \\0 & 0 & 1 & {- 1}\end{bmatrix}*\begin{bmatrix}1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & {- 1} & 0 \\0 & 0 & 0 & {- 1}\end{bmatrix}}} \\{= {0.5*\begin{bmatrix}1 & 1 & 1 & 1 \\1 & {- 1} & 1 & {- 1} \\1 & 1 & {- 1} & {- 1} \\1 & {- 1} & {- 1} & 1\end{bmatrix}}} \\{W_{2} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix}1 & 1 & 0 & 0 \\j & {- j} & 0 & 0 \\0 & 0 & 1 & 1 \\0 & 0 & j & {- j}\end{bmatrix}*\begin{bmatrix}1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & {- 1} & 0 \\0 & 0 & 0 & {- 1}\end{bmatrix}}} \\{= {0.5*\begin{bmatrix}1 & 1 & 1 & 1 \\j & {- j} & j & {- j} \\1 & 1 & {- 1} & {- 1} \\j & {- j} & {- j} & j\end{bmatrix}}} \\{W_{3} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix}1 & 1 & 0 & 0 \\1 & {- 1} & 0 & 0 \\0 & 0 & 1 & 1 \\0 & 0 & j & {- j}\end{bmatrix}*\begin{bmatrix}1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & {- 1} & 0 \\0 & 0 & 0 & {- 1}\end{bmatrix}}} \\{= {0.5*\begin{bmatrix}1 & 1 & 1 & 1 \\1 & {- 1} & j & {- j} \\1 & 1 & {- 1} & {- 1} \\1 & {- 1} & {- j} & j\end{bmatrix}}} \\{W_{4} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix}1 & 1 & 0 & 0 \\j & {- j} & 0 & 0 \\0 & 0 & 1 & 1 \\0 & 0 & 1 & {- 1}\end{bmatrix}*\begin{bmatrix}1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & {- 1} & 0 \\0 & 0 & 0 & {- 1}\end{bmatrix}}} \\{= {0.5*\begin{bmatrix}1 & 1 & 1 & 1 \\j & {- j} & 1 & {- 1} \\1 & 1 & {- 1} & {- 1} \\j & {- j} & {- 1} & 1\end{bmatrix}}} \\{W_{5} = {{{diag}( {1,1,1,{- 1}} )}*D\; F\; T}} \\{= {0.5*\begin{bmatrix}1 & 1 & 1 & 1 \\1 & j & {- 1} & {- j} \\1 & {- 1} & 1 & {- 1} \\{- 1} & j & 1 & {- j}\end{bmatrix}}} \\{W_{6} = {{{diag}( {1,\frac{( {1 + j} )}{\sqrt{2}},j,\frac{( {{- 1} + j} )}{\sqrt{2}}} )}*D\; F\; T}} \\{= {0.5*\begin{bmatrix}1 & 1 & 1 & 1 \\\frac{( {1 + j} )}{\sqrt{2}} & \frac{( {{- 1} + j} )}{\sqrt{2}} & \frac{( {{- 1} - j} )}{\sqrt{2}} & \frac{( {1 - j} )}{\sqrt{2}} \\j & {- j} & j & {- j} \\\frac{( {{- 1} + j} )}{\sqrt{2}} & \frac{( {1 + j} )}{\sqrt{2}} & \frac{( {1 - j} )}{\sqrt{2}} & \frac{( {{- 1} - j} )}{\sqrt{2}}\end{bmatrix}}}\end{matrix}$

For example, where the number of physical antennas of the base stationis four, the matrices or the codewords included in the codebook used inthe downlink of the single user MIMO communication system according tothe fourth example may be given by the following

TABLE 12 Transmit Codebook Transmission Transmission TransmissionTransmission Index Rank 1 Rank 2 Rank 3 Rank 4 1 C_(1,1) = W1(;, 2)C_(1,2) = W1(;, 1 2) C_(1,3) = W1(;, 1 2 3) C_(1,4) = W1(;, 1 2 3 4) 2C_(2,1) = W1(;, 3) C_(2,2) = W1(;, 1 3) C_(2,3) = W1(;, 1 2 4) C_(2,4) =W2(;, 1 2 3 4) 3 C_(3,1) = W1(;, 4) C_(3,2) = W1(;, 1 4) C_(3,3) = W1(;,1 3 4) C_(3,4) = W3(;, 1 2 3 4) 4 C_(4,1) = W2(;, 2) C_(4,2) = W1(;, 23) C_(4,3) = W1(;, 2 3 4) C_(4,4) = W4(;, 1 2 3 4) 5 C_(5,1) = W2(;, 3)C_(5,2) = W1(;, 2 4) C_(5,3) = W2(;, 1 2 3) C_(5,4) = W5(;, 1 2 3 4) 6C_(6,1) = W2(;, 4) C_(6,2) = W1(;, 3 4) C_(6,3) = W2(;, 1 2 4) C_(6,4) =W6(;, 1 2 3 4) 7 C_(7,1) = W3(;, 1) C_(7,2) = W2(;, 1 3) C_(7,3) = W2(;,1 3 4) n/a 8 C_(8,1) = W4(;, 1) C_(8,2) = W2(;, 1 4) C_(8,3) = W2(;, 2 34) n/a 9 C_(9,1) = W5(;, 1) C_(9,2) = W2(;, 2 3) C_(9,3) = W3(;, 1 2 3)n/a 10 C_(10,1) = W5(;, 2) C_(10,2) = W2(;, 2 4) C_(10,3) = W3(;, 1 3 4)n/a 11 C_(11,1) = W5(;, 3) C_(11,2) = W3(;, 1 3) C_(11,3) = W4(;, 1 2 3)n/a 12 C_(12,1) = W5(;, 4) C_(12,2) = W3(;, 1 4) C_(12,3) = W4(;, 1 3 4)n/a 13 C_(13,1) = W6(;, 1) C_(13,2) = W4(;, 1 3) C_(13,3) = W5(;, 1 2 3)n/a 14 C_(14,1) = W6(;, 2) C_(14,2) = W4(;, 1 4) C_(14,3) = W5(;, 1 3 4)n/a 15 C_(15,1) = W6(;, 3) C_(15,2) = W5(;, 1 3) C_(15,3) = W6(;, 1 2 4)n/a 16 C_(16,1) = W6(;, 4) C_(16,2) = W6(;, 2 4) C_(16,3) = W6(;, 2 3 4)n/a

The codewords included in the above Table 12 may be expressed asfollows:

$C_{1,1} = \begin{matrix}0.5000 \\{- 0.5000} \\0.5000 \\{- 0.5000}\end{matrix}$ $C_{2,1} = \begin{matrix}0.5000 \\0.5000 \\{- 0.5000} \\{- 0.5000}\end{matrix}$ $C_{3,1} = \begin{matrix}0.5000 \\{- 0.5000} \\{- 0.5000} \\0.5000\end{matrix}$ $C_{4,1} = \begin{matrix}0.5000 \\{0 - {0.5000i}} \\0.5000 \\{0 - {0.5000i}}\end{matrix}$

$C_{5,1} = \begin{matrix}0.5000 \\{0 + {0.5000i}} \\{- 0.5000} \\{0 - {0.5000i}}\end{matrix}$ $C_{6,1} = \begin{matrix}0.5000 \\{0 - {0.5000i}} \\{- 0.5000} \\{0 + {0.5000i}}\end{matrix}$ $C_{7,1} = \begin{matrix}0.5000 \\0.5000 \\0.5000 \\0.5000\end{matrix}$ $C_{8,1} = \begin{matrix}0.5000 \\{0 + {0.5000i}} \\0.5000 \\{0 + {0.5000i}}\end{matrix}$ $C_{9,1} = \begin{matrix}0.5000 \\0.5000 \\0.5000 \\{- 0.5000}\end{matrix}$ $C_{10,1} = \begin{matrix}0.5000 \\{0 + {0.5000i}} \\{- 0.5000} \\{0 + {0.5000i}}\end{matrix}$

$C_{11,1} = \begin{matrix}0.5000 \\{- 0.5000} \\0.5000 \\0.5000\end{matrix}$ $C_{12,1} = \begin{matrix}0.5000 \\{0 - {0.5000i}} \\{- 0.5000} \\{0 - {0.5000i}}\end{matrix}$ $C_{13,1} = \begin{matrix}0.5000 \\{0.3536 + {0.3536i}} \\{0 + {0.5000i}} \\{{- 0.3536} + {0.3536i}}\end{matrix}$ $C_{14,1} = \begin{matrix}0.5000 \\{{- 0.3536} + {0.3536i}} \\{0 - {0.5000i}} \\{0.3536 + {0.3536i}}\end{matrix}$ $C_{15,1} = \begin{matrix}0.5000 \\{{- 0.3536} - {0.3536i}} \\{0 + {0.5000i}} \\{0.3536 - {0.3536i}}\end{matrix}$ $C_{16,1} = \begin{matrix}0.5000 \\{0.3536 - {0.3536i}} \\{0 - {0.5000i}} \\{{- 0.3536} - {0.3536i}}\end{matrix}$

$C_{1,2} = \begin{matrix}0.5000 & 0.5000 \\{- 0.5000} & 0.5000 \\0.5000 & 0.5000 \\{- 0.5000} & 0.5000\end{matrix}$ $C_{2,2} = \begin{matrix}0.5000 & 0.5000 \\0.5000 & 0.5000 \\{- 0.5000} & 0.5000 \\{- 0.5000} & 0.5000\end{matrix}$ $C_{3,2} = \begin{matrix}0.5000 & 0.5000 \\{- 0.5000} & 0.5000 \\{- 0.5000} & 0.5000 \\0.5000 & 0.5000\end{matrix}$ $C_{4,2} = \begin{matrix}0.5000 & 0.5000 \\{- 0.5000} & 0.5000 \\0.5000 & {- 0.5000} \\{- 0.5000} & {- 0.5000}\end{matrix}$ $C_{5,2} = \begin{matrix}0.5000 & 0.5000 \\{- 0.5000} & {- 0.5000} \\0.5000 & {- 0.5000} \\{- 0.5000} & 0.5000\end{matrix}$ $C_{6,2} = \begin{matrix}0.5000 & 0.5000 \\0.5000 & {- 0.5000} \\{- 0.5000} & {- 0.5000} \\{- 0.5000} & 0.5000\end{matrix}$

$C_{7,2} = \begin{matrix}0.5000 & 0.5000 \\{0 + {0.5000i}} & {0 + {0.5000i}} \\{- 0.5000} & 0.5000 \\{0 - {0.5000i}} & {0 + {0.5000i}}\end{matrix}$ $C_{8,2} = \begin{matrix}0.5000 & 0.5000 \\{0 - {0.5000i}} & {0 + {0.5000i}} \\{- 0.5000} & 0.5000 \\{0 + {0.5000i}} & {0 + {0.5000i}}\end{matrix}$ $C_{9,2} = \begin{matrix}0.5000 & 0.5000 \\{0 - {0.5000i}} & {0 + {0.5000i}} \\{- 0.5000} & {- 0.5000} \\{0 - {0.5000i}} & {0 - {0.5000i}}\end{matrix}$ $C_{10,2} = \begin{matrix}0.5000 & 0.5000 \\{0 - {0.5000i}} & {0 - {0.5000i}} \\0.5000 & {- 0.5000} \\{0 - {0.5000i}} & {0 + {0.5000i}}\end{matrix}$ $C_{11,2} = \begin{matrix}0.5000 & 0.5000 \\{0 + {0.5000i}} & 0.5000 \\{- 0.5000} & 0.5000 \\{0 - {0.5000i}} & 0.5000\end{matrix}$ $C_{12,2} = \begin{matrix}0.5000 & 0.5000 \\{0 - {0.5000i}} & 0.5000 \\{- 0.5000} & 0.5000 \\{0 + {0.5000i}} & 0.5000\end{matrix}$

$C_{13,2} = \begin{matrix}0.5000 & 0.5000 \\0.5000 & {0 + {0.5000i}} \\{- 0.5000} & 0.5000 \\{- 0.5000} & {0 + {0.5000i}}\end{matrix}$ $C_{14,2} = \begin{matrix}0.5000 & 0.5000 \\{- 0.5000} & {0 + {0.5000i}} \\{- 0.5000} & 0.5000 \\0.5000 & {0 + {0.5000i}}\end{matrix}$ $C_{15,2} = \begin{matrix}0.5000 & 0.5000 \\0.5000 & {- 0.5000} \\0.5000 & 0.5000 \\{- 0.5000} & 0.5000\end{matrix}$ $C_{16,2} = \begin{matrix}0.5000 & 0.5000 \\{{- 0.3536} + {0.3536i}} & {0.3536 - {0.3536i}} \\{0 - {0.5000i}} & {0 - {0.5000i}} \\{0.3536 + {0.3536i}} & {{- 0.3536} - {0.3536i}}\end{matrix}$ $C_{1,3} = \begin{matrix}0.5000 & 0.5000 & 0.5000 \\0.5000 & {- 0.5000} & 0.5000 \\0.5000 & 0.5000 & {- 0.5000} \\0.5000 & {- 0.5000} & {- 0.5000}\end{matrix}$ $C_{2,3} = \begin{matrix}0.5000 & 0.5000 & 0.5000 \\0.5000 & {- 0.5000} & {- 0.5000} \\0.5000 & 0.5000 & {- 0.5000} \\0.5000 & {- 0.5000} & 0.5000\end{matrix}$

$C_{3,3} = \begin{matrix}0.5000 & 0.5000 & 0.5000 \\0.5000 & 0.5000 & {- 0.5000} \\0.5000 & {- 0.5000} & {- 0.5000} \\0.5000 & {- 0.5000} & 0.5000\end{matrix}$ $C_{4,3} = \begin{matrix}0.5000 & 0.5000 & 0.5000 \\{- 0.5000} & 0.5000 & {- 0.5000} \\0.5000 & {- 0.5000} & {- 0.5000} \\{- 0.5000} & {- 0.5000} & 0.5000\end{matrix}$ $C_{5,3} = \begin{matrix}0.5000 & 0.5000 & 0.5000 \\{0 + {{.05000}i}} & {0 - {0.5000\; i}} & {0 + {0.5000i}} \\0.5000 & 0.5000 & {- 0.5000} \\{0 + {0.5000i}} & {0 - {0.5000\; i}} & {0 - {0.5000i}}\end{matrix}$ $C_{6,3} = \begin{matrix}0.5000 & 0.5000 & 0.5000 \\{0 + {0.5000i}} & {0 - {0.5000i}} & {0 - {0.5000i}} \\0.5000 & 0.5000 & {- 0.5000} \\{0 + {0.5000i}} & {0 - {0.5000i}} & {0 + {0.5000i}}\end{matrix}$ $C_{7,3} = \begin{matrix}0.5000 & 0.5000 & 0.5000 \\{0 + {0.5000i}} & {0 + {0.5000i}} & {0 - {0.5000i}} \\0.5000 & {- 0.5000} & {- 0.5000} \\{0 + {0.5000i}} & {0 - {0.5000i}} & {0 + {0.5000i}}\end{matrix}$ $C_{8,3} = \begin{matrix}0.5000 & 0.5000 & 0.5000 \\{0 - {0.5000i}} & {0 + {0.5000i}} & {0 - {0.5000i}} \\0.5000 & {- 0.5000} & {- 0.5000} \\{0 - {0.5000i}} & {0 - {0.5000i}} & {0 + {0.5000i}}\end{matrix}$

$C_{9,3} = \begin{matrix}0.5000 & 0.5000 & 0.5000 \\0.5000 & {- 0.5000} & {0 + {0.5000i}} \\0.5000 & 0.5000 & {- 0.5000} \\0.5000 & {- 0.5000} & {0 - {0.5000i}}\end{matrix}$ $C_{10,3} = \begin{matrix}0.5000 & 0.5000 & 0.5000 \\0.5000 & {0 + {0.5000i}} & {0 - {0.5000i}} \\0.5000 & {- 0.5000} & {- 0.5000} \\0.5000 & {0 - {0.5000i}} & {0 + {0.5000i}}\end{matrix}$ $C_{11,3} = \begin{matrix}0.5000 & 0.5000 & 0.5000 \\{0 + {0.5000i}} & {0 - {0.5000i}} & 0.5000 \\0.5000 & 0.5000 & {- 0.5000} \\{0 + {0.5000i}} & {0 - {0.5000i}} & {- 0.5000}\end{matrix}$ $C_{12,3} = \begin{matrix}0.5000 & 0.5000 & 0.5000 \\{0 + {0.5000i}} & 0.5000 & {- 0.5000} \\0.5000 & {- 0.5000} & {- 0.5000} \\{0 + {0.5000i}} & {- 0.5000} & 0.5000\end{matrix}$ $C_{13,3} = \begin{matrix}0.5000 & 0.5000 & 0.5000 \\0.5000 & {0 + {0.5000i}} & {- 0.5000} \\0.5000 & {- 0.5000} & 0.5000 \\{- 0.5000} & {0 + {0.5000i}} & 0.5000\end{matrix}$ $C_{14,3} = \begin{matrix}0.5000 & 0.5000 & 0.5000 \\0.5000 & {- 0.5000} & {0 - {0.5000i}} \\0.5000 & 0.5000 & {- 0.5000} \\{- 0.5000} & 0.5000 & {0 - {0.5000i}}\end{matrix}$

$C_{15,3} = \begin{matrix}0.5000 & 0.5000 & 0.5000 \\{0.3536 + {0.3536i}} & {{- 0.3536} + {0.3536i}} & {0.3536 - {0.3536i}} \\{0 + {0.5000i}} & {0 - {0.5000i}} & {0 - {0.5000i}} \\{{- 0.3536} + {0.3536i}} & {0.3536 + {0.3536i}} & {{- 0.3536} - {0.3536i}}\end{matrix}$ $C_{16,3} = \begin{matrix}0.5000 & 0.5000 & 0.5000 \\{{- 0.3536} + {0.3536i}} & {{- 0.3536} - {0.3536i}} & {0.3536 - {0.3536i}} \\{0 - {0.5000i}} & {0 + {0.5000i}} & {0 - {0.5000i}} \\{0.3536 + {0.3536i}} & {0.3536 - {0.3536i}} & {{- 0.3536} - {0.3536i}}\end{matrix}$ $C_{1,4} = \begin{matrix}0.5000 & 0.5000 & 0.5000 & 0.5000 \\0.5000 & {- 0.5000} & 0.5000 & {- 0.5000} \\0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\0.5000 & {- 0.5000} & {- 0.5000} & 0.5000\end{matrix}$ $C_{2,4} = \begin{matrix}0.5000 & 0.5000 & 0.5000 & 0.5000 \\{0 + {0.5000i}} & {0 - {0.5000i}} & {0 + {0.5000i}} & {0 - {0.5000i}} \\0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\{0 + {0.5000i}} & {0 - {0.5000i}} & {0 - {0.5000i}} & {0 + {0.5000i}}\end{matrix}$ $C_{3,4} = \begin{matrix}0.5000 & 0.5000 & 0.5000 & 0.5000 \\0.5000 & {- 0.5000} & {0 + {0.5000i}} & {0 - {0.5000i}} \\0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\0.5000 & {- 0.5000} & {0 - {0.5000i}} & {0 + {0.5000i}}\end{matrix}$ $C_{4,4} = \begin{matrix}0.5000 & 0.5000 & 0.5000 & 0.5000 \\{0 + {0.5000i}} & {0 - {0.5000i}} & 0.5000 & {- 0.5000} \\0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\{0 + {0.5000i}} & {0 - {0.5000i}} & {{- 0.5000}i} & 0.5000\end{matrix}$

$C_{5,4} = \begin{matrix}0.5000 & 0.5000 & 0.5000 & 0.5000 \\0.5000 & {0 + {0.5000i}} & {- 0.5000} & {0 - {0.5000i}} \\0.5000 & {- 0.5000} & 0.5000 & {- 0.5000} \\{- 0.5000} & {0 + {0.5000i}} & {0.5000i} & {0 - {0.5000i}}\end{matrix}$ $C_{6,4} = \begin{matrix}0.5000 & 0.5000 & 0.5000 & 0.5000 \\{0.3536 + {0.3536\mspace{11mu} i}} & {{- 0.3536} + {0.3536\mspace{11mu} i}} & {{- 0.3536} - {0.3536\mspace{11mu} i}} & {0.3536 - {0.3536\mspace{11mu} i}} \\{0 + {0.5000\mspace{11mu} i}} & {0 - {0.5000\mspace{11mu} i}} & {0 + {0.5000\; i}} & {0 - {0.5000\mspace{11mu} i}} \\{{- 0.3536} + {0.3536\mspace{11mu} i}} & {0.3536 + {0.3536\mspace{11mu} i}} & {0.3536 - {0.3536\mspace{11mu} i}} & {{- 0.3536} - {0.3536\mspace{11mu} i}}\end{matrix}$

6-2) Codebook used in a downlink of a multi-user MIMO communicationsystem performing unitary precoding where a number of physical antennasof a base station is four:

For example, the codebook used in the downlink of the multi-user MIMOcommunication system according to the fourth example may be designed byappropriately combining subsets of two matrices W₃ and W₆ as follows:

$\begin{matrix}{W_{3} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix}1 & 1 & 0 & 0 \\1 & {- 1} & 0 & 0 \\0 & 0 & 1 & 1 \\0 & 0 & j & {- j}\end{bmatrix}*\begin{bmatrix}1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & {- 1} & 0 \\0 & 0 & 0 & {- 1}\end{bmatrix}}} \\{= {0.5*\begin{bmatrix}1 & 1 & 1 & 1 \\1 & {- 1} & j & {- j} \\1 & 1 & {- 1} & {- 1} \\1 & {- 1} & {- j} & j\end{bmatrix}}}\end{matrix}$ $\begin{matrix}{W_{6} = {{{diag}( {1,\frac{( {1 + j} )}{\sqrt{2}},j,\frac{( {{- 1} + j} )}{\sqrt{2}}} )}*D\; F\; T}} \\{= {0.5*\begin{bmatrix}1 & 1 & 1 & 1 \\\frac{( {1 + j} )}{\sqrt{2}} & \frac{( {{- 1} + j} )}{\sqrt{2}} & \frac{( {{- 1} - j} )}{\sqrt{2}} & \frac{( {1 - j} )}{\sqrt{2}} \\j & {- j} & j & {- j} \\\frac{( {{- 1} + j} )}{\sqrt{2}} & \frac{( {1 + j} )}{\sqrt{2}} & \frac{( {1 - j} )}{\sqrt{2}} & \frac{( {{- 1} - j} )}{\sqrt{2}}\end{bmatrix}}}\end{matrix}$

For example, where the number of physical antennas of the base stationis four, the matrices or the codewords included in the codebook used inthe downlink of the multi-user MIMO communication system according tothe fourth example may be given by the following Table 13:

codeword used at Transmit Codebook the mobile station Index forquantization 1 M₁ = W3(;, 1) 2 M₂ = W3(;, 2) 3 M₃ = W3(;, 3) 4 M₄ =W3(;, 4) 5 M₅ = W6(;, 1) 6 M₆ = W6(;, 2) 7 M₇ = W6(;, 3) 8 M₈ = W6(;, 4)

The codewords included in the above Table 13 may be expressed asfollows:

$M_{1} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\0.5000\end{matrix} \\0.5000\end{matrix} \\0.5000\end{matrix}$ $M_{2} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\{- 0.5000}\end{matrix} \\0.5000\end{matrix} \\{- 0.5000}\end{matrix}$ $M_{3} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\{0 + {0.5000i}}\end{matrix} \\{- 0.5000}\end{matrix} \\{0 - {0.5000i}}\end{matrix}$ $M_{4} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\{0 - {0.5000\; i}}\end{matrix} \\{- 0.5000}\end{matrix} \\{0 + {0.5000\; i}}\end{matrix}$

$M_{5} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\{0.3536 + {0.3536i}}\end{matrix} \\{0 + {0.5000i}}\end{matrix} \\{{- 0.3536} + {0.3536i}}\end{matrix}$ $M_{6} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\{{- 0.3536} + {0.3536i}}\end{matrix} \\{0 - {0.5000i}}\end{matrix} \\{0.3536 + {0.3536i}}\end{matrix}$ $M_{7} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\{{- 0.3536} - {0.3536i}}\end{matrix} \\{0 + {0.5000i}}\end{matrix} \\{0.3536 - {0.3536i}}\end{matrix}$ $M_{8} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\{0.3536 - {0.3536i}}\end{matrix} \\{0 - {0.5000i}}\end{matrix} \\{{- 0.3536} - {0.3536i}}\end{matrix}$

6-3) Codebook used in a downlink of a multi-user MIMO communicationsystem performing non-unitary precoding where a number of physicalantennas of a base station is four:

For example, where the number of physical antennas of the base stationis four, the codewords included in the codebook used in the downlink ofthe multi-user MIMO communication system performing non-unitaryprecoding may be given by the following Table 14:

Transmit Codebook Transmission Index Rank 1 1 C_(1,1) = W1(;, 2) 2C_(2,1) = W1(;, 3) 3 C_(3,1) = W1(;, 4) 4 C_(4,1) = W2(;, 2) 5 C_(5,1) =W2(;, 3) 6 C_(6,1) = W2(;, 4) 7 C_(7,1) = W3(;, 1) 8 C_(8,1) = W4(;, 1)9 C_(9,1) = W5(;, 1) 10 C_(10,1) = W5(;, 2) 11 C_(11,1) = W5(;, 3) 12C_(12,1) = W5(;, 4) 13 C_(13,1) = W6(;, 1) 14 C_(14,1) = W6(;, 2) 15C_(15,1) = W6(;, 3) 16 C_(16,1) = W6(;, 4)

The codewords included in the above Table may be expressed as follows:

$C_{1,1} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\{- 0.5000}\end{matrix} \\0.5000\end{matrix} \\{- 0.5000}\end{matrix}$ $C_{2,1} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\0.5000\end{matrix} \\{- 0.5000}\end{matrix} \\{- 0.5000}\end{matrix}$ $C_{3,1} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\{- 0.5000}\end{matrix} \\{- 0.5000}\end{matrix} \\0.5000\end{matrix}$

$C_{4,1} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\{0 - {0.5000i}}\end{matrix} \\0.5000\end{matrix} \\{0 - {0.5000i}}\end{matrix}$ $C_{5,1} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\{0 + {0.5000i}}\end{matrix} \\{- 0.5000}\end{matrix} \\{0 - {0.5000i}}\end{matrix}$ $C_{6,1} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\{0 - {0.5000i}}\end{matrix} \\{- 0.5000}\end{matrix} \\{0 + {0.5000i}}\end{matrix}$ $C_{7,1} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\0.5000\end{matrix} \\0.5000\end{matrix} \\0.5000\end{matrix}$ $C_{8,1} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\{0 + {0.5000i}}\end{matrix} \\0.5000\end{matrix} \\{0 + {0.5000i}}\end{matrix}$ $C_{9,1} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\0.5000\end{matrix} \\0.5000\end{matrix} \\{- 0.5000}\end{matrix}$

$C_{10,1} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\{0 + {0.5000i}}\end{matrix} \\{- 0.5000}\end{matrix} \\{0 + {0.5000i}}\end{matrix}$ $C_{11,1} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\{- 0.5000}\end{matrix} \\0.5000\end{matrix} \\0.5000\end{matrix}$ $C_{12,1} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\{0 - {0.5000i}}\end{matrix} \\{- 0.5000}\end{matrix} \\{0 - {0.5000i}}\end{matrix}$ $C_{13,1} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\{0.3536 + {0.3536i}}\end{matrix} \\{0 + {0.5000i}}\end{matrix} \\{{- 0.3536} + {0.3536i}}\end{matrix}$ $C_{14,1} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\{{- 0.3536} + {0.3536i}}\end{matrix} \\{0 - {0.5000i}}\end{matrix} \\{0.3536 + {0.3536i}}\end{matrix}$ $C_{15,1} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\{{- 0.3536} - {0.3536i}}\end{matrix} \\{0 + {0.5000i}}\end{matrix} \\{0.3536 - {0.3536i}}\end{matrix}$ $C_{16,1} = \begin{matrix}\begin{matrix}\begin{matrix}0.5000 \\{0.3536 - {0.3536i}}\end{matrix} \\{0 - {0.5000i}}\end{matrix} \\{{- 0.3536} - {0.3536i}}\end{matrix}$

Various examples of codewords according to a transmission rank and anumber of antennas in a downlink of a single user MIMO communicationsystem and a multi-user MIMO communication system have been describedabove. The aforementioned codewords may be modified using various typesof schemes or shapes, and thus are not limited to the aforementionedexamples. For example, based on the disclosures and teachings providedherein, one skilled in the art may obtain the substantially samecodebook by changing phases of columns of the aformentioned codewords,for example, by multiplying a complex exponential and the columns of thecodewords. As another example, one skilled in the art may multiply ‘−1’and the columns of the codewords.

However, a performance or properties of the codebook may not change bychanging the phases of the columns of the codewords. Accordingly, it isunderstood that a codebook generated by changing the phases of thecolumns of the codewords is the same or equivalent as a codebookincluding the original codewords prior to the changing of the phases.Also, it is understood that a codebook generated by swapping columns ofthe original codewords of a codebook is the same or equivalent as thecodebook including the original codewords prior to the swapping.Specific values that are included in the codebook generated by changingthe phases of the columns of the original codewords, or in the codebookgenerated by swapping the columns of the original codewords will beomitted herein for conciseness.

Where the number of antennas of the base station is four, theaforementioned codebooks assume that a MIMO communication systemgenerally uses a 4-bit codebook. However, embodiments are not limitedthereto. For example, where a 6-bit codebook includes codewords that arethe same as or substantially same as aforementioned codewords includedin the 4-bit codebook, it is understood that the 6-bit codebook may beequivalent to exemplary codebooks described above.

7) Codebook used in a downlink of a single user MIMO communicationsystem where a number of physical antennas of a base station is eight:

For example, where the number of physical antennas of the base stationis eight, the codebook used in the downlink of the single user MIMOcommunication system may be designed as given by the following Equation9:

$\begin{matrix}{W_{0} = {{\frac{1}{\sqrt{8}}\begin{bmatrix}1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\1 & {\mathbb{e}}^{j\;{\pi/4}} & {\mathbb{e}}^{{j\pi}/2} & {\mathbb{e}}^{j\; 3{\pi/4}} & {\mathbb{e}}^{j\pi} & {\mathbb{e}}^{j\; 5\;{\pi/4}} & {\mathbb{e}}^{j\; 3{\pi/2}} & {\mathbb{e}}^{j\; 7{\pi/4}} \\1 & {\mathbb{e}}^{{j\pi}/2} & {\mathbb{e}}^{j\;\pi} & {\mathbb{e}}^{j\; 3{\pi/2}} & 1 & {\mathbb{e}}^{{j\pi}/2} & {\mathbb{e}}^{j\pi} & {\mathbb{e}}^{j\; 3{\pi/2}} \\1 & {\mathbb{e}}^{j\; 3{\pi/4}} & {\mathbb{e}}^{j\; 3{\pi/2}} & {\mathbb{e}}^{j\;{\pi/4}} & {\mathbb{e}}^{j\pi} & {\mathbb{e}}^{j\; 7{\pi/4}} & {\mathbb{e}}^{{j\pi}/2} & {\mathbb{e}}^{j\; 5{\pi/4}} \\1 & {\mathbb{e}}^{j\pi} & 1 & {\mathbb{e}}^{j\pi} & 1 & {\mathbb{e}}^{j\pi} & 1 & {\mathbb{e}}^{j\pi} \\1 & {\mathbb{e}}^{j\; 5{\pi/4}} & {\mathbb{e}}^{{j\pi}/2} & {\mathbb{e}}^{j\; 7{\pi/4}} & {\mathbb{e}}^{j\pi} & {\mathbb{e}}^{{j\pi}/4} & {\mathbb{e}}^{j\; 3{\pi/2}} & {\mathbb{e}}^{j\; 3{\pi/4}} \\1 & {\mathbb{e}}^{j\; 6\;{\pi/4}} & {\mathbb{e}}^{j\pi} & {\mathbb{e}}^{j\;{\pi/2}} & 1 & {\mathbb{e}}^{j\; 3{\pi/2}} & {\mathbb{e}}^{j\pi} & {\mathbb{e}}^{{j\pi}/2} \\1 & {\mathbb{e}}^{j\; 7{\pi/4}} & {\mathbb{e}}^{j\; 3{\pi/2}} & {\mathbb{e}}^{j\; 5{\pi/4}} & {\mathbb{e}}^{j\pi} & {\mathbb{e}}^{j\; 3{\pi/4}} & {\mathbb{e}}^{{j\pi}/2} & {\mathbb{e}}^{{j\pi}/4}\end{bmatrix}}.}} & (9)\end{matrix}$

Matrices included in the codebook for the single user MIMO communicationsystem may be determined as given by the following Table 15:

Transmit Codebook Transmission Transmission Transmission TransmissionIndex Rank 1 Rank 2 Rank 3 Rank 4 1 C1, 1 = W0(;, 1) C1, 2 = W0(;, 1 2)C1, 3 = W0(;, 1 2 3) C1, 4 = W0(;, 1 2 3 4) 2 C2, 1 = W0(;, 2) C2, 2 =W0(;, 3 4) C2, 3 = W0(;, 3 4 5) C2, 4 = W0(;, 3 4 5 6) 3 C3, 1 = W0(;,3) C3, 2 = W0(;, 5 6) C3, 3 = W0(;, 5 6 7) C3, 4 = W0(;, 5 6 7 8) 4 C4,1 = W0(;, 4) C4, 2 = W0(;, 7 8) C4, 3 = W0(;, 7 8 1) C4, 4 = W0(;, 7 8 12) 5 C5, 1 = W0(;, 5) C5, 2 = W0(;, 1 3) C5, 3 = W0(;, 1 3 5) C5, 4 =W0(;, 1 3 5 7) 6 C6, 1 = W0(;, 6) C6, 2 = W0(;, 2 4) C6, 3 = W0(;, 2 46) C6, 4 = W0(;, 2 4 6 8) 7 C7, 1 = W0(;, 7) C7, 2 = W0(;, 5 7) C7, 3 =W0(;, 5 7 1) C7, 4 = W0(;, 5 7 1 4) 8 C8, 1 = W0(;, 8) C8, 2 = W0(;, 68) C8, 3 = W0(;, 6 8 2) C8, 4 = W0(;, 6 8 2 3)

Referring to the above Table 15, where the transmission rank is 4, theprecoding matrix may be generated based on any one of W₀(;,1 2 3 4),W₀(;,3 4 5 6), W₀(;,5 6 7 8), W₀(;,7 8 1 2), W₀(;,1 3 5 7), W₀(;,2 4 68), W₀(;,5 7 1 4), and W₀(;,6 8 2 3). Here, W_(k)(;,n m o p) denotes amatrix that includes an n^(th) column vector, an m^(th) column vector,an o^(th) column vector, and a p^(th) column vector of W_(k).

Where the transmission rank is 3, the precoding matrix may be generatedbased on any one of W₀(;,1 2 3), W₀(;,3 4 5), W₀(;,5 6 7), W₀(;,7 8 1),W₀(;,1 3 5), W₀(;,2 4 6), W₀(;,5 7 1), and W₀(;,6 8 2). Here, W_(k)(;,nm o) denotes a matrix that includes the n^(th) column vector, the m^(th)column vector, and the o^(th) column vector of W_(k).

Where the transmission rank is 2, the precoding matrix may be generatedbased on any one of W₀(;,1 2), W₀(;,3 4), W₀(;,5 6), W₀(;,7 8), W₀(;,13), W₀(;,2 4), W₀(;,5 7), and W₀(;,6 8). Here, W_(k)(;,n m) denotes amatrix that includes the n^(th) column vector and the m^(th) columnvector of W_(k).

Where the transmission rank is 1, the precoding matrix may be generatedbased on any one of W₀(;,1), W₀(;,2), W₀(;,3), W₀(;,4), W₀(;,5),W₀(;,6), W₀(;,7), and W₀(;,8). Here, W_(k)(;,n) denotes the n^(th)column vector of W_(k).

Updated Codebook

A MIMO communication system according to an exemplary embodiment mayupdate a codebook according to a time correlation coefficient (p) of achannel that is formed between at least one user and a base station.

In an environment where a channel varies over time, it may beinappropriate to use a fixed codebook. The MIMO communication system maydetect the change in the channel to thereby adaptively update thecodebook.

Generally, the channel may be modeled as given by the following Equation10:H _(τ) =ρ·H _(τ-1)+Δ  (10),where H_(τ) denotes a channel vector or a channel matrix in a τ^(th)time instance, ρ denotes the time correlation coefficient greater than 0and less than 1, and Δ denotes complex noise, and has a normaldistribution where the average of Δ is zero and the variance of Δ is1−ρ.

In a MIMO communication system according to an exemplary embodiment, aterminal may calculate the time correlation coefficient (ρ) according tothe above Equation 10. The terminal may feed back the calculated timecorrelation coefficient (ρ) to the base station. The base station mayadaptively update the codebook based on the fed back time correlationcoefficient (ρ).

Here, it is assumed that the codebook before the codebook is updated is{θ}={Θ₁, . . . , Θ₂ _(B) } and the updated codebook is {{tilde over(θ)}}={{tilde over (Θ)}₁, . . . , {tilde over (Θ)}₂ _(B) }. {tilde over(Θ)}_(i) denotes an i^(th) element of the updated codebook and B denotesa number of feedback bits.

Where a precoding matrix used in a (τ−1)^(th) time instance is F_(τ-1),the base station according to an aspect may calculate a new precodingmatrix F_(τ) using {tilde over (Θ)}_(i). {tilde over (Θ)}_(i) is anelement of the updated codebook. F₀ corresponding to an initial value ofF_(τ) may be the aforementioned precoding matrix with respect to two,four, and eight transmit antennas. Specifically, F_(τ) may berepresented using the previous precoding matrix F_(τ-1) and the element{tilde over (Θ)}_(i) of the updated codebook, as given by the followingEquation 11:F _(τ)={tilde over (Θ)}_(i) F _(τ-1)  (11).

In a multi-user MIMO communication system, F_(τ) may be a precodingmatrix for a single user. Accordingly, F_(τ) may be different for eachuser. The base station may generate the precoding matrix by schedulingactive terminals to transmit data and by combining F_(τ) correspondingto each user.

1) Design of {θ}={Θ₁, . . . , Θ₂ _(B) }:

For example, elements or codewords of {θ}={Θ₁, . . . , Θ₂ _(B) } havethe dimension of N_(t)×N_(t). Each of the elements may be a unitarymatrix. {θ}={Θ₁, . . . , Θ₂ _(B) } may be equally spaced.

Design schemes of {θ}={Θ₁, . . . , Θ₂ _(B) } may vary, however twoexemplary schemes will be described here.

(1) Full unitary matrices {θ}={Θ₁, . . . , Θ₂ _(B) } including fullunitary matrices may be proposed according to the following Equation 12:

$\begin{matrix}{{{\Theta_{l} = {\Phi^{l}D}},{l = 1},\ldots\mspace{11mu},2^{B}}{\Phi = \begin{bmatrix}{\mathbb{e}}^{j\; 2\;{{\pi\phi}_{1}/2^{B}}} & 0 & \cdots & 0 \\0 & {\mathbb{e}}^{j\; 2{{\pi\phi}_{2}/2^{B}}} & \cdots & 0 \\\vdots & \vdots & \ddots & 0 \\0 & 0 & 0 & {\mathbb{e}}^{j\; 2\;{{\pi\phi}_{N_{i}}/2^{B}}}\end{bmatrix}}{{\phi_{i} \in \{ {1,2,\ldots\mspace{11mu},2^{B}} \}},}} & (12)\end{matrix}$

where D denotes a DFT matrix with the dimension of N_(t)×N_(t).

For example, where N_(t)=4, [φ₁,φ₂,φ₃,φ₄] with respect to the number offeedback bits B may be determined as given by the following Table 16:

B [φ₁, φ₂, φ₃, φ₄] B = 2 [1, 2, 3, 4] B = 3 [2, 4, 5, 6] B = 4 [1, 3, 4,8] B = 5 [11, 18, 22, 23]

For example, where B=3 and N_(t)=2, {θ}={Θ₁, . . . , Θ₂ _(B) } may begiven by the following Table 17:

θ₁ $\lfloor \begin{matrix}{{- 0.0136} + {0.6753i}} & {0.7368 - {0.0288i}} \\{{- 0.7276} + {0.1198i}} & {{- 0.1489} - {0.6588i}}\end{matrix} \rfloor\quad$ θ₂ $\lfloor \begin{matrix}{{- 0.6021} + {0.6871i}} & {0.0081 + {0.4065i}} \\{{- 0.0729} + {0.4000i}} & {{- 0.4847} - {0.7744i}}\end{matrix} \rfloor\quad$ θ₃ ${\begin{matrix}{{- 0.0877} - {0.9095i}} & {0.3790 + {0.1464i}} \\{{- 0.3929} - {0.1035i}} & {{- 0.6041} + {0.6856i}}\end{matrix}}\quad$ θ₄ $\lfloor \begin{matrix}{{- 0.7424} + {0.3706i}} & {0.2022 - {0.5202i}} \\{0.3856 - {0.4035i}} & {0.0213 - {0.8295i}}\end{matrix} \rfloor\quad$ θ₅ $\lfloor \begin{matrix}{{- 0.2839} + {0.0675i}} & {{- 0.7744} + {0.5614i}} \\{0.6104 + {0.7363i}} & {{- 0.2582} - {0.1359i}}\end{matrix} \rfloor\quad$ θ₆ $\lfloor \begin{matrix}{{- 0.4786} + {0.1916i}} & {0.8547 + {0.0614i}} \\{{- 0.0827} + {0.8529i}} & {{- 0.2692} + {0.4397i}}\end{matrix} \rfloor\quad$ θ₇ $\lfloor \begin{matrix}{{- 0.1309} - {0.8846i}} & {{- 0.4081} - {0.1838i}} \\{0.1516 + {0.4211i}} & {{- 0.8718} - {0.1992i}}\end{matrix} \rfloor\quad$ θ₈ $\lfloor \begin{matrix}{{- 0.0707} + {0.9650i}} & {{- 0.0601} - {0.2455i}} \\{{- 0.1422} + {0.2090i}} & {{- 0.2711} + {0.9288i}}\end{matrix} \rfloor\quad$

(2) Diagonal unitary matrices {θ}={Θ₁, . . . , Θ₂ _(B) } includingdiagonal unitary matrices may be proposed according to the followingEquation 13:Θ_(l)=Φ^(l) ,l=1, . . . ,2^(B)Φ=diag[e ^(j2πφ) ¹ ^(/2) ^(B) e ^(j2πφ) ² ^(/2) ^(B) . . . e ^(jφN) ^(t)^(/2) ^(B) ]φ_(i)ε{1,2,3, . . . ,2^(B)}  (13).

Referring to the above Equation 13, {θ}={Θ₁, . . . , Θ₂ _(B) } may bedesigned using diagonal matrices. For example, where N_(t)=4,[φ₁,φ₂,φ₃,φ₄] with respect to the number of feedback bits B may bedetermined as given by the following Table 18:

B [φ₁, φ₂, φ₃, φ₄] B = 2 [1, 2, 3, 4] B = 3 [2, 4, 5, 6] B = 4 [1, 3, 4,8] B = 5 [11, 18, 22, 23]

2) Calculation of {{tilde over (θ)}}={{tilde over (Θ)}₁, . . . , {tildeover (Θ)}₂ _(B) }:

In a MIMO communication system according to an exemplary embodiment, abase station or user terminals may update {θ}={Θ₁, . . . , Θ₂ _(B) }according to a time correlation coefficient (ρ) of a channel to therebygenerate a new codebook {{tilde over (θ)}}={{tilde over (Θ)}₁, . . . ,{tilde over (Θ)}₂ _(B) }.

With respect to i=1, 2, 3 . . . 2^(B), Ψ_(i)(ρ,Θ_(i))=ρI+√{square rootover (1−ρ²)}Θ_(i) may be calculated. In this instance, the updatedcodebook may be provided using various types of schemes. Hereinafter,two exemplary schemes will be described. Here, I denotes an identitymatrix.

(1) First Scheme:

In the first scheme, the updated codebook {{tilde over (θ)}}={{tildeover (Θ)}₁, . . . , {tilde over (Θ)}₂ _(B) } according to an exemplaryembodiment may be calculated using the following Equation 14:

$\begin{matrix}{{\overset{\sim}{\Theta}}_{i} = {\underset{{\overset{\sim}{\Theta}}_{i}}{\arg\;\min}{{{{\Psi_{i}( {\rho,\Theta_{i}} )} - {\overset{\sim}{\Theta}}_{i}}}_{F}.}}} & (14)\end{matrix}$

In the above Equation 14, ∥x∥_(F) denotes a Frobenious norm of x.

Here, it is assumed that where a singular value decomposition (SVD) isperformed for Ψ_(i)(ρ,Θ_(i))=ρI+√{square root over (1−ρ²)}Θ_(i),Ψ_(i)(ρ,Θ_(i))=Φ_(i)Λ_(i)B_(i)*. In this instance, a solution of theabove Equation 14 may be calculated using the following Equation 15:{tilde over (Θ)}_(i)=Φ_(i) B _(i)*  (15).

In the first scheme, the updated codebook {θ}={Θ₁, . . . , Θ₂ _(B) } maybe calculated according to the above Equation 14 or the above equation15.

(2) Second Scheme:

In the second scheme, the updated codebook {{tilde over (θ)}}={{tildeover (Θ)}₁, . . . , {tilde over (Θ)}₂ _(B) } according to an exemplaryembodiment may be calculated using the following Equation 16:{tilde over(Θ)}_(i)=[Ψ_(i)(ρ,Θ_(i))*Ψ_(i)(ρ,Θ_(i))]^(−1/2)Ψ_(i)(ρ,Θ_(i))  (16).

The updated codebook {{tilde over (θ)}}={{tilde over (Θ)}₁, . . . ,{tilde over (Θ)}₂ _(B) } calculated according to the first scheme or thesecond scheme may be dynamically calculated based on the timecorrelation coefficient.

The time correlation coefficient may be quantized. Updated codebookscorresponding to the quantized values may be stored in a memory. In thisinstance, the updated codebooks may have no need to be dynamicallycalculated. Specifically, where the terminal feeds back the timecorrelation coefficient to the base station, the base station may selectany one codebook from a plurality of updated codebooks that are storedin the memory.

Various Examples of the Updated Codebook {{tilde over (θ)}}={{tilde over(Θ)}₁, . . . , {tilde over (Θ)}₂ _(B) }

As described above, the updated codebook {{tilde over (θ)}}={{tilde over(Θ)}₁, . . . , {tilde over (Θ)}₂ _(B) } may be calculated differentlyaccording to various parameters. Hereinafter, an example of {θ}={Θ₁, . .. , Θ₂ _(B) } depending on the number of feedback bits, {θ}={Θ₁, . . . ,Θ₂ _(B) }, or design schemes will be described.

1) Where N_(t)=4, the number of feedback bits B=4, {θ}={Θ₁, . . . , Θ₂_(B) } includes full unitary matrices, and the first scheme, that is,the above Equation 14 or the above Equation 15 is used, examples of thecodebook that {{tilde over (θ)}}={{tilde over (Θ)}₁, . . . , {tilde over(Θ)}₂ _(B) } is updated according to the time correlation coefficientmay follow as:

(1) Time correlation coefficient=0

${\overset{\sim}{\Theta}}_{1} = \begin{matrix}{0.4619 + {0.1913i}} & {0.4619 + {0.1913i}} & {0.4619 + {0.1913i}} & {0.4619 + {0.1913i}} \\{0.1913 + {0.4619i}} & {{- 0.4619} + {0.1913i}} & {{- 0.1913} - {0.4619i}} & {0.4619 - {0.1913i}} \\{0.0000 + {0.5000i}} & {{- 0.0000} - {0.5000i}} & {0.0000 + {0.5000i}} & {{- 0.0000} - {0.5000i}} \\{{- 0.5000} + {0.0000i}} & {0.0000 + {0.5000i}} & {0.5000 - {0.0000i}} & {{- 0.0000} - {0.5000i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{2} = \begin{matrix}{0.3536 + {0.3536i}} & {0.3536 + {0.3536i}} & {0.3536 + {0.3536i}} & {0.3536 + {0.3536i}} \\{{- 0.3536} + {0.3536i}} & {{- 0.3536} - {0.3536i}} & {0.3536 - {0.3536i}} & {0.3536 + {0.3536i}} \\{{- 0.5000} + {0.000i}} & {0.5000 - {0.0000i}} & {{- 0.5000} + {0.0000i}} & {0.5000 - {0.0000i}} \\{0.5000 - {0.0000i}} & {{- 0.0000} - {0.5000i}} & {{- 0.5000} + {0.0000i}} & {0.0000 + {0.5000i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{3} = \begin{matrix}{0.1913 + {0.4619i}} & {0.1913 + {0.4619i}} & {0.1913 + {0.4619i}} & {0.1913 + {0.4619i}} \\{{- 0.4619} - {0.1913i}} & {0.1913 - {0.4619i}} & {0.4619 + {0.1913i}} & {{- 0.1913} + {0.4619i}} \\{{- 0.0000} - {0.5000i}} & {0.0000 + {0.5000i}} & {{- 0.0000} - {0.5000i}} & {0.0000 + {0.5000i}} \\{{- 0.5000} + {0.0000i}} & {0.0000 + {0.5000i}} & {0.5000 - {0.0000i}} & {{- 0.0000} - {0.5000i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{4} = \begin{matrix}{{- 0.0000} + {0.5000i}} & {{- 0.0000} + {0.5000i}} & {{- 0.0000} + {0.5000i}} & {{- 0.0000} + {0.5000i}} \\{{- 0.0000} - {0.5000i}} & {0.5000 - {0.0000i}} & {0.0000 + {0.5000i}} & {{- 0.5000} + {0.0000i}} \\{0.5000 - {0.0000i}} & {{- 0.5000} + {0.0000i}} & {0.5000 - {0.0000i}} & {{- 0.5000} + {0.0000i}} \\{0.5000 - {0.0000i}} & {{- 0.0000} - {0.5000i}} & {{- 0.5000} + {0.0000i}} & {0.0000 + {0.5000i}}\end{matrix}$

${\overset{\sim}{\Theta}}_{5} = \begin{matrix}{{- 0.1913} + {0.4619i}} & {{- 0.1913} + {0.4619i}} & {{- 0.1913} + {0.4619i}} & {{- 0.1913} + {0.4619i}} \\{0.4619 - {0.1913i}} & {0.1913 + {0.4619i}} & {{- 0.4619} + {0.1913i}} & {{- 0.1913} - {0.4619i}} \\{0.0000 + {0.5000i}} & {{- 0.0000} - {0.5000i}} & {0.0000 + {0.5000i}} & {{- 0.0000} - {0.5000i}} \\{{- 0.5000} + {0.0000i}} & {0.0000 + {0.5000i}} & {0.5000 - {0.0000i}} & {{- 0.0000} - {0.5000i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{6} = \begin{matrix}{{- 0.3536} + {0.3536i}} & {{- 0.3536} + {0.3536i}} & {{- 0.3536} + {0.3536i}} & {{- 0.3536} + {0.3536i}} \\{0.3536 + {0.3536i}} & {{- 0.3536} + {0.3536i}} & {{- 0.3536} - {0.3536i}} & {0.3536 - {0.3536i}} \\{{- 0.5000} + {0.0000i}} & {0.5000 - {0.0000i}} & {{- 0.5000} + {0.000i}} & {0.5000 - {0.0000i}} \\{0.5000 - {0.0000i}} & {{- 0.0000} - {0.5000i}} & {{- 0.5000} + {0.0000i}} & {0.0000 + {0.5000i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{7} = \begin{matrix}{{- 0.4619} + {0.1913i}} & {{- 0.4619} + {0.1913i}} & {{- 0.4619} + {0.1913i}} & {{- 0.4619} + {0.1913i}} \\{{- 0.1913} + {0.4619i}} & {{- 0.4619} - {0.1913i}} & {0.1913 - {0.4619i}} & {0.4619 + {0.1913i}} \\{{- 0.0000} - {0.5000i}} & {0.0000 + {0.5000i}} & {{- 0.0000} - {0.5000i}} & {0.0000 + {0.5000i}} \\{{- 0.5000} + {0.0000i}} & {0.0000 + {0.5000i}} & {0.5000 - {0.0000i}} & {{- 0.0000} - {0.5000i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{8} = \begin{matrix}{{- 0.5000} - {0.0000i}} & {{- 0.5000} - {0.0000i}} & {{- 0.5000} - {0.0000i}} & {{- 0.5000} - {0.0000i}} \\{{- 0.5000} + {0.0000i}} & {{- 0.0000} - {0.5000i}} & {0.5000 - {0.0000i}} & {0.0000 + {0.5000i}} \\{0.5000 - {0.0000i}} & {{- 0.5000} + {0.0000i}} & {0.5000 - {0.0000i}} & {{- 0.5000} + {0.0000i}} \\{0.5000 - {0.0000i}} & {{- 0.0000} - {0.5000i}} & {{- 0.5000} + {0.0000i}} & {0.0000 + {0.5000i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{9} = \begin{matrix}{{- 0.4619} - {0.1913i}} & {{- 0.4619} - {0.1913i}} & {{- 0.4619} - {0.1913i}} & {{- 0.4619} - {0.1913i}} \\{{- 0.1913} - {0.4619i}} & {0.4619 - {0.1913i}} & {0.1913 + {0.4619i}} & {{- 0.4619} + {0.1913i}} \\{0.0000 + {0.5000i}} & {{- 0.0000} - {0.5000i}} & {0.0000 + {0.5000i}} & {{- 0.0000} - {0.5000i}} \\{{- 0.5000} + {0.0000i}} & {0.0000 + {0.5000i}} & {0.5000 - {0.0000i}} & {{- 0.0000} - {0.5000i}}\end{matrix}$

${\overset{\sim}{\Theta}}_{10} = \begin{matrix}{{- 0.3536} - {0.3536i}} & {{- 0.3536} - {0.3536i}} & {{- 0.3536} - {0.3536i}} & {{- 0.3536} - {0.3536i}} \\{0.3536 - {0.3536\; i}} & {0.3536 + {0.3536i}} & {{- 0.3536} + {0.3536i}} & {{- 0.3536} - {0.3536i}} \\{{- 0.5000} + {0.0000i}} & {0.5000 - {0.0000i}} & {{- 0.5000} + {0.0000i}} & {0.5000 - {0.0000i}} \\{0.5000 - {0.0000i}} & {{- 0.0000} - {0.5000i}} & {{- 0.5000} + {0.0000i}} & {0.0000 + {0.5000i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{11} = \begin{matrix}{{- 0.1913} - {0.4619i}} & {{- 0.1913} - {0.4619i}} & {{- 0.1913} - {0.4619i}} & {{- 0.1913} - {0.4619i}} \\{0.4619 + {0.1913i}} & {{- 0.1913} + {0.4619i}} & {{- 0.4619} - {0.1913i}} & {0.1913 - {0.4619i}} \\{{- 0.0000} - {0.5000i}} & {0.0000 + {0.5000i}} & {{- 0.0000} - {0.5000i}} & {0.0000 + {0.5000i}} \\{{- 0.5000} + {0.000i}} & {0.0000 + {0.5000i}} & {0.5000 - {0.0000i}} & {{- 0.0000} - {0.5000i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{12} = \begin{matrix}{0.000 - {0.5000i}} & {0.0000 - {0.5000i}} & {0.0000 - {0.5000i}} & {0.0000 - {0.5000i}} \\{0.0000 + {0.5000i}} & {{- 0.5000} + {0.0000i}} & {{- 0.0000} - {0.5000i}} & {0.5000 - {0.0000i}} \\{0.5000 - {0.0000i}} & {{- 0.5000} + {0.0000i}} & {0.5000 - {0.0000i}} & {{- 0.5000} + {0.0000i}} \\{0.5000 - {0.0000i}} & {{- 0.0000} - {0.5000i}} & {{- 0.5000} + {0.0000i}} & {0.0000 + {0.5000i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{13} = \begin{matrix}{0.1913 - {0.4619i}} & {0.1913 - {0.4619i}} & {0.1913 - {0.4619i}} & {0.1913 - {0.4619i}} \\{{- 0.4619} + {0.1913i}} & {{- 0.1913} - {0.4619i}} & {0.4619 - {0.1913i}} & {0.1913 + {0.4619i}} \\{0.0000 + {0.5000i}} & {{- 0.0000} - {0.5000i}} & {0.0000 + {0.5000i}} & {{- 0.0000} - {0.5000i}} \\{{- 0.5000} + {0.0000i}} & {0.0000 + {0.5000i}} & {0.5000 - {0.0000i}} & {{- 0.0000} - {0.5000i}}\end{matrix}$ $\begin{matrix}{{\overset{\sim}{\Theta}}_{14} = {0.3536 - {0.3536i}}} & {0.3536 - {0.356i}} & {0.3536 - {0.3536i}} & {0.3536 - {0.3536i}} \\{{- 0.3536} - {0.3536i}} & {0.3536 - {0.3536i}} & {0.3536 + {0.3536i}} & {{- 0.3536} + {0.3536i}} \\{{- 0.5000} + {0.0000i}} & {0.5000 - {0.0000i}} & {{- 0.5000} + {0.0000i}} & {0.5000 - {0.0000i}} \\{0.5000 - {0.0000i}} & {{- 0.0000} - {0.5000i}} & {{- 0.5000} + {0.0000i}} & {0.0000 + {0.5000i}}\end{matrix}$

${\overset{\sim}{\Theta}}_{15} = \begin{matrix}{0.4619 - {0.1913i}} & {0.4619 - {0.1913i}} & {0.4619 - {0.1913i}} & {0.4619 - {0.1913i}} \\{0.1913 - {0.4619i}} & {0.4619 + {0.1913i}} & {{- 0.1913} + {0.4619i}} & {{- 0.4619} - {0.1913i}} \\{{- 0.0000} - {0.5000i}} & {0.0000 + {0.5000i}} & {{- 0.0000} - {0.5000i}} & {0.0000 + {0.5000i}} \\{{- 0.5000} + {0.0000i}} & {0.0000 + {0.5000i}} & {0.5000 - {0.0000i}} & {{- 0.000} - {0.5000i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{16} = \begin{matrix}{0.5000 + {0.0000i}} & {0.5000 + {0.0000i}} & {0.5000 + {0.0000i}} & {0.5000 + {0.0000i}} \\{0.5000 - {0.0000i}} & {0.0000 + {0.5000i}} & {{- 0.5000} + {0.0000i}} & {{- 0.0000} - {0.5000i}} \\{0.5000 - {0.0000i}} & {{- 0.5000} + {0.0000i}} & {0.5000 - {0.0000i}} & {{- 0.5000} + {0.0000i}} \\{0.5000 - {0.0000i}} & {{- 0.0000} - {0.5000i}} & {{- 0.5000} + {0.0000i}} & {0.0000 + {0.5000i}}\end{matrix}$

(2) Time correlation coefficient=0.7

${\overset{\sim}{\Theta}}_{1} = \begin{matrix}{0.8240 + {0.0189i}} & {0.2994 + {0.2817i}} & {0.2696 + {0.1981i}} & {0.1678 + {0.1077i}} \\{0.0126 + {0.4109i}} & {0.4062 - {0.2560i}} & {{- 0.2069} - {0.5350i}} & {0.5164 + {0.0693i}} \\{{- 0.0798} + {0.3249i}} & {0.0136 - {0.5734i}} & {0.6644 + {0.2768i}} & {0.0282 - {0.2007i}} \\{{- 0.1963} - {0.0353i}} & {{- 0.2616} + {0.4506i}} & {0.2007 + {0.0282i}} & {0.6376 - {0.4911i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{2} = \begin{matrix}{0.7988 + {0.2608i}} & {0.2153 + {0.2412i}} & {0.3405 + {0.2145i}} & {0.1516 + {0.0666i}} \\{{- 0.2412} + {0.2153i}} & {0.2857 - {0.6645i}} & {0.5114 + {0.1398i}} & {0.0769 + {0.2920i}} \\{{- 0.3925} + {0.0891i}} & {0.2627 + {0.4605i}} & {0.2717 - {0.2708i}} & {0.6304 - {0.1112i}} \\{0.1543 - {0.0601i}} & {0.1521 - {0.2609i}} & {{- 0.6304} + {0.1112i}} & {0.6077 + {0.3198i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{3} = \begin{matrix}{0.7348 + {0.4182i}} & {0.2456 + {0.2255i}} & {0.0941 + {0.1479i}} & {0.2562 + {0.2788i}} \\{{- 0.3331} + {0.0142i}} & {0.7508 - {0.2853i}} & {0.2798 + {0.2541i}} & {{- 0.1102} + {0.2979i}} \\{{- 0.0303} - {0.1726i}} & {{- 0.1277} + {0.3558i}} & {0.6625 - {0.4782i}} & {{- 0.1775} + {0.3567i}} \\{{- 0.3557} + {0.1300i}} & {0.0122 + {0.3174i}} & {0.3567 + {0.1775i}} & {0.6808 - {0.3653i}}\end{matrix}$

${\overset{\sim}{\Theta}}_{4} = \begin{matrix}{0.5441 + {0.0085i}} & {{- 0.2136} + {0.2732i}} & {{- 0.1350} + {0.4619i}} & {{- 0.1789} + {0.5657i}} \\{0.2136 - {0.2732i}} & {0.7635 - {0.1612i}} & {{- 0.1287} + {0.1506i}} & {{- 0.4619} - {0.1350i}} \\{0.4619 + {0.1350i}} & {{- 0.1506} - {0.1287i}} & {0.7635 - {0.1612i}} & {{- 0.2737} - {0.2136i}} \\{0.5657 + {0.1789i}} & {0.1350 - {0.4619i}} & {{- 0.2732} - {0.2136i}} & {0.5441 + {0.0085i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{5} = \begin{matrix}{0.4414 + {0.5844i}} & {0.1239 + {0.2899i}} & {0.0638 + {0.2469i}} & {{- 0.0778} + {0.5415i}} \\{0.1174 - {0.2926i}} & {0.5415 + {0.4973i}} & {{- 0.1298} + {0.2643i}} & {{- 0.4031} - {0.3331i}} \\{0.1535 + {0.2037i}} & {{- 0.1945} - {0.2210i}} & {0.5783 + {0.5124i}} & {{- 0.0782} - {0.4951i}} \\{{- 0.5300} + {0.1353i}} & {0.2449 + {0.4620i}} & {0.4951 - {0.0782i}} & {0.4191 + {0.0201i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{6} = \begin{matrix}{0.4758 + {0.5513i}} & {{- 0.2076} + {0.2408i}} & {{- 0.0712} + {0.5514i}} & {{- 0.1783} + {0.1662i}} \\{0.2408 + {0.2076i}} & {0.5396 + {0.4524i}} & {{- 0.3868} - {0.2803i}} & {0.2279 - {0.3505i}} \\{{- 0.4402} + {0.3396i}} & {0.4717 - {0.0753i}} & {0.3841 + {0.4175i}} & {0.3750 + {0.0139i}} \\{0.2436 + {0.0086i}} & {{- 0.0867} - {0.4090i}} & {{- 0.3750} - {0.0139i}} & {0.6905 + {0.3849i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{7} = \begin{matrix}{0.4042 + {0.4158i}} & {{- 0.3971} + {0.5424i}} & {{- 0.3672} - {0.0638i}} & {{- 0.1717} + {0.2082i}} \\{0.1028 + {0.6644i}} & {0.4264 - {0.1701i}} & {0.2303 - {0.1613i}} & {0.3462 + {0.3721i}} \\{0.1995 - {0.3148i}} & {{- 0.1510} + {0.2371i}} & {0.6448 - {0.5523i}} & {{- 0.0875} + {0.2314i}} \\{{- 0.2383} + {0.1267i}} & {{- 0.2113} + {0.4622i}} & {0.2314 + {0.0875i}} & {0.6554 - {0.4221i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{8} = \begin{matrix}{0.4041 - {0.2684i}} & {{- 0.2633} - {0.3293i}} & {{- 0.4370} + {0.0141i}} & {{- 0.6264} + {0.0574i}} \\{{- 0.2633} - {0.3293i}} & {0.6101 - {0.5655i}} & {0.2477 + {0.0292i}} & {{- 0.1400} + {0.2200i}} \\{0.4370 - {0.0141i}} & {{- 0.2477} - {0.0292i}} & {0.8442 + {0.0116i}} & {{- 0.1768} + {0.0493i}} \\{0.6264 - {0.0574i}} & {0.1400 - {0.2200i}} & {{- 0.1768} + {0.0493i}} & {0.6383 + {0.3086i}}\end{matrix}$

${\overset{\sim}{\Theta}}_{9} = \begin{matrix}{0.3810 - {0.5349i}} & {{- 0.3185} - {0.3493i}} & {{- 0.3206} + {0.0369i}} & {{- 0.1642} - {0.4627i}} \\{0.0218 - {0.4722i}} & {0.8072 - {0.1391i}} & {0.1984 + {0.1319i}} & {{- 0.2124} + {0.0617i}} \\{0.1568 + {0.2821i}} & {{- 0.1328} - {0.1978i}} & {0.6503 + {0.5464i}} & {{- 0.1313} - {0.3167i}} \\{{- 0.3288} - {0.3647i}} & {{- 0.0242} + {0.2198i}} & {0.3167 - {0.1313i}} & {0.6245 - {0.4500i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{10} = \begin{matrix}{0.4747 - {0.5481i}} & {{- 0.1729} - {0.2064i}} & {{- 0.0676} - {0.5598i}} & {{- 0.2185} - {0.1898i}} \\{0.2064 - {0.1729i}} & {0.8108 + {0.1633i}} & {{- 0.3141} + {0.2256i}} & {{- 0.1773} - {0.2500i}} \\{{- 0.4437} - {0.3480i}} & {0.3816 + {0.0626i}} & {0.3891 - {0.3976i}} & {0.4714 - {0.0288i}} \\{0.2887 - {0.0203i}} & {0.0514 - {0.3021i}} & {{- 0.4714} + {0.0288i}} & {0.6901 + {0.3508i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{11} = \begin{matrix}{0.5936 - {0.3846i}} & {{- 0.3426} - {0.2840i}} & {{- 0.0888} - {0.3966i}} & {0.0617 - {0.3642i}} \\{0.4431 - {0.0414i}} & {0.5668 + {0.5554i}} & {{- 0.3627} + {0.0575i}} & {0.1153 - {0.1550i}} \\{0.0698 - {0.4004i}} & {0.1919 + {0.3131i}} & {0.6732 - {0.3571i}} & {{- 0.1491} + {0.3114i}} \\{{- 0.3128} - {0.1964i}} & {{- 0.0472} + {0.1874i}} & {0.3114 + {0.1491i}} & {0.6509 - {0.5323i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{12} = \begin{matrix}{0.5830 - {0.5182i}} & {0.4165 - {0.3063i}} & {0.0851 - {0.1869i}} & {{- 0.0688} - {0.2782i}} \\{{- 0.4165} + {0.3063i}} & {0.3110 - {0.4163i}} & {0.2059 - {0.48840i}} & {0.3694 - {0.2226i}} \\{0.1869 + {0.0851i}} & {{- 0.4840} - {0.2059i}} & {0.7985 - {0.0734i}} & {{- 0.1812} + {0.0736i}} \\{0.2782 - {0.0688i}} & {{- 0.2226} - {0.3694i}} & {{- 0.1812} + {0.0736i}} & {0.6517 + {0.5186i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{13} = \begin{matrix}{0.7342 - {0.4210i}} & {0.2750 - {0.2455i}} & {0.0978 - {0.1472i}} & {0.2285 - {0.2536i}} \\{{- 0.3680} - {0.0208i}} & {0.6070 - {0.3975i}} & {0.3063 - {0.2895i}} & {0.0993 + {0.3876i}} \\{{- 0.0341} + {0.1734i}} & {{- 0.1503} - {0.3937i}} & {0.6631 + {0.4741i}} & {{- 0.1501} - {0.3228i}} \\{{- 0.3218} - {0.1140i}} & {{- 0.0566} + {0.3960i}} & {0.3228 - {0.1501i}} & {0.6838 - {0.3593i}}\end{matrix}$

${\overset{\sim}{\Theta}}_{14} = \begin{matrix}{0.7953 - {0.2408i}} & {0.1812 - {0.1376i}} & {0.3861 - {0.2588i}} & {0.1125 - {0.1704i}} \\{{- 0.1376} - {0.1812i}} & {0.7724 - {0.3493i}} & {0.0589 + {0.3942i}} & {{- 0.2518} + {0.0856i}} \\{{- 0.4561} - {0.0900i}} & {0.3204 + {0.2371i}} & {0.4276 - {0.2914i}} & {0.5958 - {0.0473i}} \\{0.2000 - {0.0409i}} & {0.1176 - {0.2386i}} & {{- 0.5958} + {0.0473i}} & {0.6547 + {0.3190i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{15} = \begin{matrix}{0.8388 - {0.0681i}} & {0.2026 - {0.2099i}} & {0.2129 - {0.1422i}} & {0.3609 - {0.1043i}} \\{{- 0.0052} - {0.2917i}} & {0.8336 + {0.1865i}} & {{- 0.1005} + {0.3624i}} & {{- 0.2077} - {0.0248i}} \\{{- 0.0499} - {0.2511i}} & {0.0458 + {0.3733i}} & {0.6797 - {0.3936i}} & {{- 0.1876} + {0.3753i}} \\{{- 0.3734} - {0.0417i}} & {0.0565 + {0.2013i}} & {0.3753 + {0.1876i}} & {0.6730 - {0.4314i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{16} = \begin{matrix}{0.5000 + {0.0000i}} & {0.5000 - {0.0000i}} & {0.5000 - {0.0000i}} & {0.5000 - {0.0000i}} \\{0.5000 - {0.0000i}} & {0.3500 + {0.3571i}} & {{- 0.5000} + {0.0000i}} & {{- 0.3500} - {0.3571i}} \\{0.5000 - {0.0000i}} & {{- 0.5000} + {0.0000i}} & {0.5000 + {0.0000i}} & {{- 0.5000} + {0.0000i}} \\{0.5000 - {0.0000i}} & {{- 0.3500} - {0.3571i}} & {{- 0.5000} + {0.0000i}} & {0.3500 + {0.3571i}}\end{matrix}$

(3) Time correlation coefficient=0.75

${\overset{\sim}{\Theta}}_{1} = \begin{matrix}{0.8690 + {0.0214i}} & {0.2188 + {0.2754i}} & {0.2210 + {0.1977i}} & {0.1689 + {0.0650i}} \\{{- 0.0401} + {0.3494i}} & {0.6156 - {0.2142i}} & {{- 0.1296} - {0.4964i}} & {0.4124 + {0.1349i}} \\{{- 0.0981} + {0.2799i}} & {0.0702 - {0.5082i}} & {0.7459 + {0.2616i}} & {{- 0.0157} - {0.1546i}} \\{{- 0.1809} + {0.0046i}} & {{- 0.2824} + {0.3294i}} & {0.1546 - {0.0157i}} & {0.7464 - {0.4447i}}\end{matrix}$ ${{\overset{\sim}{\Theta}}_{2} = \begin{matrix}{0.8359 + {0.2414i}} & {0.2181 + {0.2150i}} & {0.3079 + {0.1975i}} & {0.1146 + {0.0485i}} \\{{- 0.2150} + {0.2181i}} & {0.8518 - {0.4014i}} & {{- 0.0143} + {0.1187i}} & {0.0719 + {0.0049i}} \\{{- 0.3574} + {0.0780i}} & {{- 0.0941} + {0.0738i}} & {0.7736 - {0.0724i}} & {0.4889 + {0.0959i}} \\{0.1153 - {0.0467i}} & {{- 0.0474} - {0.0543i}} & {{- 0.4889} - {0.0959i}} & {0.7826 + {0.3443i}}\end{matrix}}\;$

${\overset{\sim}{\Theta}}_{3} = \begin{matrix}{0.7851 + {0.3865i}} & {0.2318 + {0.1968i}} & {0.0834 + {0.1175i}} & {0.2421 + {0.2497i}} \\{{- 0.3030} + {0.0247i}} & {0.7946 - {0.2645i}} & {0.2495 + {0.2409i}} & {{- 0.0922} + {0.2782i}} \\{{- 0.0320} - {0.1405i}} & {{- 0.1271} + {0.3227i}} & {0.7299 - {0.4414i}} & {{- 0.1761} + {0.3168i}} \\{{- 0.3233} + {0.1282i}} & {0.0213 + {0.2923i}} & {0.3168 + {0.1761i}} & {0.7399 - {0.3380i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{4} = \begin{matrix}{0.8392 + {0.0971i}} & {{- 0.1312} + {0.0624i}} & {{- 0.2104} + {0.2486i}} & {{- 0.2774} + {0.2867i}} \\{0.1312 - {0.0624i}} & {0.9258 - {0.0994i}} & {0.0168 + {0.0746i}} & {{- 0.2486} - {0.2104i}} \\{0.2486 + {0.2104i}} & {{- 0.0746} + {0.0168i}} & {0.9258 - {0.0994i}} & {{- 0.0624} - {0.1312i}} \\{0.2867 + {0.2774i}} & {0.2104 - {0.2486i}} & {{- 0.0624} - {0.1312i}} & {0.8392 + {0.0971i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{5} = \begin{matrix}{0.8073 + {0.4105i}} & {{- 0.1364} + {0.0270i}} & {{- 0.1749} + {0.2852i}} & {0.1428 + {0.1674i}} \\{0.1155 + {0.0774i}} & {0.9269 + {0.3117i}} & {{- 0.0698} - {0.0265i}} & {{- 0.1307} + {0.0420i}} \\{{- 0.0525} + {0.3304i}} & {0.0512 - {0.0543i}} & {0.8138 + {0.3954i}} & {{- 0.1867} - {0.1703i}} \\{{- 0.1000} + {0.1961i}} & {0.1369 + {0.0112i}} & {0.1703 - {0.1867i}} & {0.9156 - {0.1746i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{6} = \begin{matrix}{0.6506 + {0.4917i}} & {{- 0.2158} + {0.2078i}} & {0.0451 + {0.4404i}} & {{- 0.1767} + {0.1340i}} \\{0.2078 + {0.2158i}} & {0.6392 + {0.4148i}} & {{- 0.3692} - {0.2118i}} & {0.2015 - {0.3284i}} \\{{- 0.2795} + {0.3433i}} & {0.4108 - {0.1113i}} & {0.6123 + {0.3594i}} & {0.3446 - {0.0010i}} \\{0.2197 + {0.0302i}} & {{- 0.0897} - {0.3747i}} & {{- 0.3446} + {0.0010i}} & {0.7463 + {0.3558i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{7} = \begin{matrix}{0.5863 + {0.3742i}} & {{- 0.3179} + {0.5101i}} & {{- 0.3244} - {0.0767i}} & {{- 0.1087} + {0.1791i}} \\{0.1360 + {0.5855i}} & {0.5870 - {0.1596i}} & {0.2021 - {0.1067i}} & {0.3167 + {0.3408i}} \\{0.1950 - {0.2703i}} & {{- 0.1458} + {0.1759i}} & {0.7256 - {0.5082i}} & {{- 0.0862} + {0.2110i}} \\{{- 0.1690} + {0.1238i}} & {{- 0.1937} + {0.4230i}} & {0.2110 + {0.0862i}} & {0.7312 - {0.3912i}}\end{matrix}$

${{\overset{\sim}{\Theta}}_{8} = \begin{matrix}{0.6000 - {0.2303i}} & {{- 0.1410} - {0.2886i}} & {{- 0.4007} - {0.0035i}} & {{- 0.5686} + {0.0001i}} \\{{- 0.1410} - {0.2886i}} & {0.7408 - {0.5103i}} & {0.2329 + {0.0072i}} & {{- 0.1207} + {0.1373i}} \\{0.4007 + {0.0035i}} & {{- 0.2329} - {0.0072i}} & {0.8746 + {0.0112i}} & {{- 0.1337} + {0.0471i}} \\{0.5686 - {0.0001i}} & {0.1207 - {0.1373i}} & {{- 0.1337} + {0.0471i}} & {0.7338 + {0.2912i}}\end{matrix}}\;$ ${\overset{\sim}{\Theta}}_{9} = \begin{matrix}{0.6029 - {0.4728i}} & {{- 0.2392} - {0.3363i}} & {{- 0.2845} + {0.0504i}} & {{- 0.0609} - {0.3942i}} \\{0.0686 - {0.4070i}} & {0.8593 - {0.1259i}} & {0.1782 + {0.1000i}} & {{- 0.1608} + {0.0880i}} \\{0.1555 + {0.2435i}} & {{- 0.1263} - {0.1606i}} & {0.7248 + {0.5036i}} & {{- 0.1354} - {0.2783i}} \\{{- 0.2071} - {0.3409i}} & {0.0198 + {0.1823i}} & {0.2783 - {0.1354i}} & {0.7385 - {0.4076i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{10} = \begin{matrix}{0.6506 - {0.4888i}} & {{- 0.1775} - {0.1796i}} & {0.0470 - {0.4493i}} & {{- 0.2186} - {0.1485i}} \\{0.1796 - {0.1775i}} & {0.8443 + {0.1525i}} & {{- 0.2983} + {0.1797i}} & {{- 0.1531} - {0.2353i}} \\{{- 0.2845} - {0.3510i}} & {0.3380 + {0.0838i}} & {0.6092 - {0.3430i}} & {0.4310 + {0.0001i}} \\{0.2596 - {0.0496i}} & {0.0582 - {0.2747i}} & {{- 0.4310} - {0.0001i}} & {0.7479 + {0.3259i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{11} = \begin{matrix}{0.6775 - {0.3558i}} & {{- 0.3232} - {0.2420i}} & {{- 0.0886} - {0.3653i}} & {0.0747 - {0.3231i}} \\{0.3997 - {0.0574i}} & {0.6662 + {0.5107i}} & {{- 0.3204} + {0.0728i}} & {0.1017 - {0.1186i}} \\{0.0580 - {0.3714i}} & {0.1899 + {0.2681i}} & {0.7362 - {0.3312i}} & {{- 0.1466} + {0.2785i}} \\{{- 0.2699} - {0.1927i}} & {{- 0.0486} + {0.1485i}} & {0.2785 + {0.1466i}} & {0.7250 - {0.4908i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{12} = \begin{matrix}{0.7791 - {0.4435i}} & {0.3342 - {0.0845i}} & {{- 0.0087} - {0.1186i}} & {{- 0.0987} - {0.2315i}} \\{{- 0.3342} + {0.0845i}} & {0.6693 - {0.3162i}} & {0.2554 - {0.3374i}} & {0.3444 - {0.1886i}} \\{0.1186 - {0.0087i}} & {{- 0.3374} - {0.2554i}} & {0.8842 - {0.0536i}} & {{- 0.1304} + {0.0715i}} \\{0.2315 - {0.0987i}} & {{- 0.1886} - {0.3444i}} & {{- 0.1304} + {0.0715i}} & {0.7284 + {0.4794i}}\end{matrix}$

${\overset{\sim}{\Theta}}_{13} = \begin{matrix}{0.7850 - {0.3891i}} & {0.2597 - {0.2125i}} & {0.0868 - {0.1169i}} & {0.2156 - {0.2283i}} \\{{- 0.3339} - {0.0334i}} & {0.6826 - {0.3675i}} & {0.2731 - {0.2737i}} & {0.0922 + {0.3584i}} \\{{- 0.0345} + {0.1412i}} & {{- 0.1484} - {0.3571i}} & {0.7300 + {0.4377i}} & {{- 0.1498} - {0.2870i}} \\{{- 0.2934} - {0.1119i}} & {{- 0.0519} + {0.3664i}} & {0.2870 - {0.1498i}} & {0.7408 - {0.3329i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{14} = \begin{matrix}{0.8365 - {0.2221i}} & {0.1578 - {0.1409i}} & {0.3648 - {0.2126i}} & {0.0768 - {0.1482i}} \\{{- 0.1409} - {0.1578i}} & {0.8295 - {0.3201i}} & {0.0078 + {0.3392i}} & {{- 0.2173} + {0.0483i}} \\{{- 0.4083} - {0.1076i}} & {0.2454 + {0.2344i}} & {0.5836 - {0.2602i}} & {0.5461 - {0.0121i}} \\{0.1591 - {0.0504i}} & {0.1195 - {0.1878i}} & {{- 0.5461} + {0.0121i}} & {0.7319 + {0.2976i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{15} = \begin{matrix}{0.8661 - {0.0634i}} & {0.1783 - {0.1981i}} & {0.1869 - {0.1391i}} & {0.3348 - {0.0924i}} \\{{- 0.0140} - {0.2661i}} & {0.8627 + {0.1724i}} & {{- 0.0912} + {0.3357i}} & {{- 0.1839} - {0.0098i}} \\{{- 0.0570} - {0.2259i}} & {0.0442 + {0.3451i}} & {0.7395 - {0.3638i}} & {{- 0.1868} + {0.3326i}} \\{{- 0.3447} - {0.0427i}} & {0.0613 + {0.1737i}} & {0.3326 + {0.1868i}} & {0.7356 - {0.3985i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{16} = \begin{matrix}{1.0000 + {0.0000i}} & {{- 0.0000} - {0.0000i}} & {0.0000 - {0.0000i}} & {{- 0.0000} - {0.0000i}} \\{0.0000 - {0.0000i}} & {0.8750 + {0.3307i}} & {{- 0.0000} + {0.0000i}} & {0.1250 - {0.3307i}} \\{{- 0.0000} - {0.0000i}} & {0.0000 + {0.0000i}} & {1.0000 + {0.0000i}} & {0.0000 + {0.0000i}} \\{0.0000 - {0.0000i}} & {0.1250 - {0.3307i}} & {{- 0.0000} + {0.0000i}} & {0.8750 + {0.3307i}}\end{matrix}$

(4) Time correlation coefficient=0.8

${\overset{\sim}{\Theta}}_{1} = \begin{matrix}{0.9061 + {0.0305i}} & {0.1493 + {0.2433i}} & {0.1747 + {0.1833i}} & {0.1762 + {0.0374i}} \\{{- 0.0665} + {0.2776i}} & {0.7742 - {0.1303i}} & {{- 0.0710} - {0.4276i}} & {0.3006 + {0.1544i}} \\{{- {.1025}} + {0.2315i}} & {0.0980 - {0.4222i}} & {0.8156 + {0.2516i}} & {{- 0.0559} - {0.1279i}} \\{{- 0.1771} + {0.0329i}} & {{- 0.2577} + {0.2187i}} & {0.1279 - {0.0559i}} & {0.8324 - {0.3755i}}\end{matrix}$

${{\overset{\sim}{\Theta}}_{2} = \begin{matrix}{0.8719 + {0.2173i}} & {0.2025 + {0.1950i}} & {0.2785 + {0.1658i}} & {0.0839 + {0.0389i}} \\{{- 0.1950} + {0.2025i}} & {0.9136 - {0.2637i}} & {{- 0.0693} + {0.0285i}} & {0.1003 - {0.0342i}} \\{{- 0.3142} + {0.0797i}} & {{- 0.0692} - {0.0288i}} & {0.8520 + {0.0152i}} & {0.3953 + {0.0831i}} \\{0.0868 - {0.0318i}} & {{- 0.0951} - {0.0467i}} & {{- 0.3953} - {0.0831i}} & {0.8408 + {0.3317i}}\end{matrix}}\;$ ${\overset{\sim}{\Theta}}_{3} = \begin{matrix}{0.8347 + {0.3475i}} & {0.2127 + {0.1663i}} & {0.0722 + {0.0879i}} & {0.2225 + {0.2171i}} \\{{- 0.2680} + {0.0329i}} & {0.8381 - {0.2409i}} & {0.2158 + {0.2220i}} & {{- 0.0742} + {0.2555i}} \\{{- 0.0331} - {0.1089i}} & {{- 0.1225} + {0.2844i}} & {0.7956 - {0.3950i}} & {{- 0.1683} + {0.2717i}} \\{{- 0.2857} + {0.1225i}} & {0.0292 + {0.2645i}} & {0.2717 + {0.1683i}} & {0.7982 - {0.3056i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{4} = \begin{matrix}{0.9121 + {0.1709i}} & {{- 0.0579} + {0.0181i}} & {{- 0.1731} + {0.1580i}} & {{- 0.2253} + {0.1716i}} \\{0.0579 - {0.0181i}} & {0.9634 - {0.0444i}} & {0.0684 + {0.0811i}} & {{- 0.1580} - {0.1731i}} \\{0.1580 + {0.1731i}} & {{- 0.0811} + {0.0684i}} & {0.9634 - {0.0444i}} & {{- 0.0181} - {0.0579i}} \\{0.1716 + {0.2253i}} & {0.1731 - {0.1580i}} & {{- 0.0181} - {0.0579i}} & {0.9121 + {0.1709i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{5} = \begin{matrix}{0.8538 + {0.3377i}} & {{- 0.1584} + {0.0349i}} & {{- 0.1532} + {0.2848i}} & {0.1122 + {0.1161i}} \\{0.1367 + {0.0873i}} & {0.9471 + {0.2461i}} & {{- 0.0829} - {0.0467i}} & {{- 0.0751} + {0.0368i}} \\{{- 0.0325} + {0.3217i}} & {0.0749 - {0.0587i}} & {0.8560 + {0.3376i}} & {{- 0.1516} - {0.1290i}} \\{{- 0.0643} + {0.1481i}} & {0.0834 - {0.0053i}} & {0.1290 - {0.1516i}} & {0.9401 - {0.2089i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{6} = \begin{matrix}{0.7827 + {0.3953i}} & {{- 0.2172} + {0.1880i}} & {0.0981 + {0.3107i}} & {{- 0.1753} + {0.1085i}} \\{0.1880 + {0.2172i}} & {0.7316 + {0.3629i}} & {{- 0.3274} - {0.1530i}} & {0.1718 - {0.3006i}} \\{{- 0.1503} + {0.2890i}} & {0.3397 - {0.1233i}} & {0.7784 + {0.2530i}} & {0.3055 - {0.0093i}} \\{0.2007 + {0.0472i}} & {{- 0.0911} - {0.3340i}} & {{- 0.3055} + {0.0093i}} & {0.8010 + {0.3203i}}\end{matrix}$

${{\overset{\sim}{\Theta}}_{7} = \begin{matrix}{0.7427 + {0.3047i}} & {{- 0.2373} + {0.4414i}} & {{- 0.2724} - {0.0816i}} & {{- 0.0575} + {0.1423i}} \\{0.1443 + {0.4799i}} & {0.7294 - {0.1502i}} & {0.1663 - {0.0600i}} & {0.2742 + {0.2963i}} \\{0.1796 - {0.2204i}} & {{- 0.1307} + {0.1190i}} & {0.8017 - {0.4491i}} & {{- 0.0856} + {0.1900i}} \\{{- 0.1076} + {0.1094i}} & {{- 0.1688} + {0.3667i}} & {0.1900 + {0.0856i}} & {0.8024 - {0.3553i}}\end{matrix}}\;$ ${\overset{\sim}{\Theta}}_{8} = \begin{matrix}{0.7539 - {0.1575i}} & {{- 0.0536} - {0.2148i}} & {{- 0.3518} - {0.0133i}} & {{- 0.4823} - {0.0343i}} \\{{- 0.0536} - {0.2148i}} & {0.8418 - {0.4250i}} & {0.2212 - {0.0077i}} & {{- 0.0843} + {0.0746i}} \\{0.3518 + {0.0133i}} & {{- 0.2212} + {0.0077i}} & {0.9031 + {0.0111i}} & {{- 0.0956} + {0.0463i}} \\{0.4823 + {0.0343i}} & {0.0843 - {0.0746i}} & {{- 0.0956} + {0.0463i}} & {0.8152 + {0.2786i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{9} = \begin{matrix}{0.7749 - {0.3638i}} & {{- 0.1686} - {0.2951i}} & {{- 0.2431} + {0.0595i}} & {0.0045 - {0.2984i}} \\{0.0894 - {0.3279i}} & {0.9034 - {0.1063i}} & {0.1533 + {0.0703i}} & {{- 0.1249} + {0.1137i}} \\{0.1480 + {0.2018i}} & {{- 0.1147} - {0.1237i}} & {0.7967 + {0.4475i}} & {{- 0.1340} - {0.2365i}} \\{{- 0.1101} - {0.2774i}} & {0.0573 + {0.1589i}} & {0.2365 - {0.1340i}} & {0.8308 - {0.3439i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{10} = \begin{matrix}{0.7840 - {0.3922i}} & {{- 0.1783} - {0.1615i}} & {0.1003 - {0.3192i}} & {{- 0.2183} - {0.1185i}} \\{0.1615 - {0.1783i}} & {0.8762 + {0.1420i}} & {{- 0.2677} + {0.1389i}} & {{- 0.1293} - {0.2157i}} \\{{- 0.1548} - {0.2966i}} & {0.2875 + {0.0911i}} & {0.7726 - {0.2428i}} & {0.3756 + {0.0169i}} \\{0.2382 - {0.0705i}} & {0.0611 - {0.2439i}} & {{- 0.3756} - {0.0167i}} & {0.8030 + {0.2983i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{11} = \begin{matrix}{0.7587 - {0.3211i}} & {{- 0.2921} - {0.1972i}} & {{- 0.0856} - {0.3275i}} & {0.0816 - {0.2752i}} \\{0.3460 - {0.0672i}} & {0.7606 + {0.4499i}} & {{- 0.2714} + {0.0813i}} & {0.0872 - {0.0839i}} \\{0.0463 - {0.3353i}} & {0.1789 + {0.2196i}} & {0.7978 - {0.3017i}} & {{- 0.1393} + {0.2412i}} \\{{- 0.2231} - {0.1807i}} & {{- 0.0484} + {0.1109i}} & {0.2412 + {0.1393i}} & {0.7966 - {0.4369i}}\end{matrix}$

${\overset{\sim}{\Theta}}_{12} = \begin{matrix}{0.8790 - {0.3390i}} & {0.2102 + {0.0057i}} & {{- 0.0699} - {0.1039i}} & {{- 0.1152} - {0.1981i}} \\{{- 0.2102} - {0.0057i}} & {0.8450 - {0.1652i}} & {0.2257 - {0.2291i}} & {0.2922 - {0.1605i}} \\{0.1039 - {0.0699i}} & {{- 0.2291} - {0.2257i}} & {0.9312 - {0.0238i}} & {{- 0.0920} + {0.0687i}} \\{0.1981 - {0.1152i}} & {{- 0.1605} - {0.2922i}} & {{- 0.0920} + {0.0687i}} & {0.7995 + {0.4289i}}\end{matrix}$ ${{\overset{\sim}{\Theta}}_{13} = \begin{matrix}{0.8349 - {0.3496i}} & {0.2377 - {0.1773i}} & {0.0750 - {0.0873i}} & {0.1987 - {0.2001i}} \\{{- 0.2934} - {0.0427i}} & {0.7567 - {0.3309i}} & {0.2358 - {0.2508i}} & {0.0836 + {0.3232i}} \\{{- 0.0359} + {0.1094i}} & {{- 0.1414} - {0.3138i}} & {0.7954 + {0.3920i}} & {{- 0.1444} - {0.2471i}} \\{{- 0.2609} - {0.1070i}} & {{- 0.0464} + {0.3305i}} & {0.2471 - {0.1444i}} & {0.7973 - {0.3024i}}\end{matrix}}\;$ ${\overset{\sim}{\Theta}}_{14} = \begin{matrix}{0.8755 - {0.1985i}} & {0.1391 - {0.1435i}} & {0.3282 - {0.1666i}} & {0.0437 - {0.1291i}} \\{{- 0.1435} - {0.1391i}} & {0.8807 - {0.2794i}} & {{- 0.0274} + {0.2719i}} & {{- 0.1769} + {0.0193i}} \\{{- 0.3499} - {0.1143i}} & {0.1729 + {0.2116i}} & {0.7221 - {0.2043i}} & {0.4759 + {0.0145i}} \\{0.1222 - {0.0604i}} & {0.1114 - {0.1387i}} & {{- 0.4759} - {0.0145i}} & {0.8040 + {0.2769i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{15} = \begin{matrix}{0.8932 - {0.0588i}} & {0.1525 - {0.1826i}} & {0.1603 - {0.1344i}} & {0.3032 - {0.0797i}} \\{{- 0.0213} - {0.2370i}} & {0.8916 + {0.1554i}} & {{- 0.0808} + {0.3037i}} & {{- 0.1595} + {0.0042i}} \\{{- 0.0628} - {0.1995i}} & {0.0416 + {0.3115i}} & {0.7983 - {0.3274i}} & {{- 0.1792} + {0.2846i}} \\{{- 0.3106} - {0.0424i}} & {0.0649 + {0.1458i}} & {0.2846 + {0.1792i}} & {0.7970 - {0.3576i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{16} = \begin{matrix}{1.0000 + {0.0000i}} & {{- 0.0000} - {0.0000i}} & {0.0000 - {0.0000i}} & {{- 0.0000} - {0.0000i}} \\{0.0000 - {0.0000i}} & {0.9000 + {0.3000i}} & {{- 0.0000} + {0.0000i}} & {0.1000 - {0.3000i}} \\{{- 0.0000} - {0.0000i}} & {0.0000 + {0.0000i}} & {1.0000 + {0.0000i}} & {0.0000 + {0.0000i}} \\{0.0000 - {0.0000i}} & {0.1000 - {0.3000i}} & {{- 0.0000} + {0.0000i}} & {0.9000 + {0.3000i}}\end{matrix}$

(5) Time correlation coefficient=0.85

${\overset{\sim}{\Theta}}_{1} = \begin{matrix}{0.9341 + {0.0385i}} & {0.1010 + {0.2000i}} & {0.1363 + {0.1602i}} & {0.1759 + {0.0248i}} \\{{- 0.0700} + {0.2129i}} & {0.8726 - {0.0511i}} & {{- 0.0394} - {0.3477i}} & {0.2098 + {0.1391i}} \\{{- 0.0958} + {0.1873i}} & {0.0966 - {0.3363i}} & {0.8715 + {0.2348i}} & {{- 0.0787} - {0.1118i}} \\{{- 0.1720} + {0.0444i}} & {{- 0.2088} + {0.1406i}} & {0.1118 - {0.0787i}} & {0.8915 - {0.3028i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{2} = \begin{matrix}{0.9066 + {0.1882i}} & {0.1796 + {0.1707i}} & {0.2440 + {0.1328i}} & {0.0542 + {0.0339i}} \\{{- 0.1707} + {0.1796i}} & {0.9393 - {0.2014i}} & {{- 0.0669} - {0.0075i}} & {0.0976 - {0.0408i}} \\{{- 0.2665} + {0.0786i}} & {{- 0.0420} - {0.0526i}} & {0.8983 + {0.0333i}} & {0.3270 + {0.0571i}} \\{0.0623 - {0.0143i}} & {{- 0.0978} - {0.0401i}} & {{- 0.3270} - {0.0571i}} & {0.8883 + {0.2923i}}\end{matrix}$ ${{\overset{\sim}{\Theta}}_{3} = \begin{matrix}{0.8824 + {0.2999i}} & {0.1875 + {0.1338i}} & {0.0605 + {0.0599i}} & {0.1964 + {0.1805i}} \\{{- 0.2272} + {0.0379i}} & {0.8808 - {0.2134i}} & {0.1787 + {0.1963i}} & {{- 0.0571} + {0.2208i}} \\{{- 0.0329} - {0.0785i}} & {{- 0.1129} + {0.2402i}} & {0.8577 - {0.3379i}} & {{- 0.1528} + {0.2217i}} \\{{- 0.2419} + {0.1123i}} & {0.0345 + {0.2325i}} & {0.2217 + {0.1528i}} & {0.5844 - {0.2666i}}\end{matrix}}\;$ ${\overset{\sim}{\Theta}}_{4} = \begin{matrix}{0.9402 + {0.1846i}} & {{- 0.0267} + {0.0077i}} & {{- 0.1395} + {0.1192i}} & {{- 0.1782} + {0.1257i}} \\{0.0267 - {0.0077i}} & {0.9758 - {0.0208i}} & {0.0784 + {0.0819i}} & {{- 0.1192} - {0.1395i}} \\{0.1192 + {0.1395i}} & {{- 0.0819} + {0.0784i}} & {0.9758 - {0.0208i}} & {{- 0.0077} - {0.0267i}} \\{0.1257 + {0.1782i}} & {0.1395 - {0.1192i}} & {{- 0.0077} - {0.0267i}} & {0.9402 + {0.1846i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{5} = \begin{matrix}{0.8928 + {0.2846i}} & {{- 0.1482} + {0.0375i}} & {{- 0.1213} + {0.2579i}} & {0.0913 + {0.0949i}} \\{0.1313 + {0.0782i}} & {0.9596 + {0.2077i}} & {{- 0.0799} - {0.0488i}} & {{- 0.0545} + {0.0291i}} \\{{- 0.0134} + {0.2846i}} & {0.0757 - {0.0551i}} & {0.8936 + {0.2888i}} & {{- 0.1275} - {0.1084i}} \\{{- 0.0528} + {0.1207i}} & {0.0614 - {0.0060i}} & {0.1084 - {0.1275i}} & {0.9544 - {0.1998i}}\end{matrix}$

${{\overset{\sim}{\Theta}}_{6} = \begin{matrix}{0.8675 + {0.2988i}} & {{- 0.2010} + {0.1717i}} & {0.1003 + {0.2058i}} & {{- 0.1676} + {0.0884i}} \\{0.1717 + {0.2010i}} & {0.8142 + {0.3010i}} & {{- 0.2740} - {0.1115i}} & {0.1386 - {0.2641i}} \\{{- 0.0746} + {0.2164i}} & {0.2726 - {0.1149i}} & {0.8762 + {0.1535i}} & {0.2622 - {0.0104i}} \\{0.1810 + {0.0561i}} & {{- 0.0887} - {0.2848i}} & {{- 0.2622} + {0.0104i}} & {0.8538 + {0.2779i}}\end{matrix}}\;$ ${\overset{\sim}{\Theta}}_{7} = \begin{matrix}{0.8573 + {0.2257i}} & {{- 0.1676} + {0.3493i}} & {{- 0.2151} - {0.0787i}} & {{- 0.0231} + {0.1052i}} \\{0.1284 + {0.3655i}} & {0.8405 - {0.1350i}} & {0.1263 - {0.0258i}} & {0.2213 + {0.2443i}} \\{0.1550 - {0.1686i}} & {{- 0.1068} + {0.0722i}} & {0.8688 - {0.3764i}} & {{- 0.0841} + {0.1655i}} \\{{- 0.0616} + {0.0883i}} & {{- 0.1410} + {0.2979i}} & {0.1655 + {0.0841i}} & {0.8654 - {0.3106i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{8} = \begin{matrix}{0.8559 - {0.0878i}} & {{- 0.0092} - {0.1402i}} & {{- 0.2963} - {0.0139i}} & {{- 0.3876} - {0.0415i}} \\{{- 0.0092} - {0.1402i}} & {0.9066 - {0.3353i}} & {0.2051 - {0.0136i}} & {{- 0.0489} + {0.0367i}} \\{0.2963 + {0.0139i}} & {{- 0.2051} + {0.0136i}} & {0.9294 + {0.0100i}} & {{- 0.0643} + {0.0424i}} \\{0.3876 + {0.0415i}} & {0.0489 - {0.0367i}} & {{- 0.0643} + {0.0424i}} & {0.8787 + {0.2574i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{9} = \begin{matrix}{0.8825 - {0.2527i}} & {{- 0.1175} - {0.2395i}} & {{- 0.1973} + {0.0629i}} & {0.0295 - {0.2061i}} \\{0.0863 - {0.2524i}} & {0.9370 - {0.0866i}} & {0.1241 + {0.0443i}} & {{- 0.1010} + {0.1260i}} \\{0.1336 + {0.1582i}} & {{- 0.0977} - {0.0884i}} & {0.8627 + {0.3779i}} & {{- 0.1255} - {0.1919i}} \\{{- 0.0517} - {0.2017i}} & {0.0778 + {0.1415i}} & {0.1919 - {0.1255i}} & {0.8952 - {0.2769i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{10} = \begin{matrix}{0.8691 - {0.2954i}} & {{- 0.1678} - {0.1464i}} & {0.1028 - {0.2120i}} & {{- 0.2068} - {0.0972i}} \\{0.1464 - {0.1678i}} & {0.9064 + {0.1296i}} & {{- 0.2292} + {0.1074i}} & {{- 0.1064} - {0.1915i}} \\{{- 0.0772} - {0.2226i}} & {0.2380 + {0.0862i}} & {0.8714 - {0.1478i}} & {0.3144 + {0.0203i}} \\{0.2149 - {0.0775i}} & {0.0601 - {0.2107i}} & {{- 0.3144} - {0.0203i}} & {0.8549 + {0.2640i}}\end{matrix}$

${\overset{\sim}{\Theta}}_{11} = \begin{matrix}{0.8340 - {0.2790i}} & {{- 0.2492} - {0.1516i}} & {{- 0.0781} - {0.2825i}} & {0.0809 - {0.2215i}} \\{0.2834 - {0.0691i}} & {0.8444 + {0.3743i}} & {{- 0.2174} + {0.0813i}} & {0.0717 - {0.0530i}} \\{0.0360 - {0.2908i}} & {0.1583 + {0.1697i}} & {0.8564 - {0.2666i}} & {{- 0.1265} + {0.1993i}} \\{{- 0.1737} - {0.1595i}} & {{- 0.0459} + {0.0764i}} & {0.1993 + {0.1265i}} & {0.8624 - {0.3701i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{12} = \begin{matrix}{0.9257 - {0.2621i}} & {0.1257 + {0.0242i}} & {{- 0.0905} - {0.0973i}} & {{- 0.1150} - {0.1648i}} \\{{- 0.1257} - {0.0242i}} & {0.9189 - {0.0744i}} & {0.1796 - {0.1674i}} & {0.2323 - {0.1396i}} \\{0.0973 - {0.0905i}} & {{- 0.1674} - {0.1796i}} & {0.9565 - {0.0072i}} & {{- 0.0618} + {0.0585i}} \\{0.1648 - {0.1150i}} & {{- 0.1396} - {0.2323i}} & {{- 0.0618} + {0.0585i}} & {0.8637 + {0.3646i}}\end{matrix}$ ${{\overset{\sim}{\Theta}}_{13} = \begin{matrix}{0.8828 - {0.3013i}} & {0.2078 - {0.1404i}} & {0.0625 - {0.0595i}} & {0.1766 - {0.1684i}} \\{{- 0.2462} - {0.0477i}} & {0.8277 - {0.2864i}} & {0.1942 - {0.2195i}} & {0.0726 + {0.2806i}} \\{{- 0.0350} + {0.0789i}} & {{- 0.1285} - {0.2634i}} & {0.8573 + {0.3358i}} & {{- 0.1329} - {0.2032i}} \\{{- 0.2232} - {0.0987i}} & {{- 0.0403} + {0.2870i}} & {0.2032 - {0.1329i}} & {0.8525 - {0.2660i}}\end{matrix}}\;$ ${\overset{\sim}{\Theta}}_{14} = \begin{matrix}{0.9108 - {0.1712i}} & {0.1237 - 0.14041} & {0.2787 - 0.1260} & {0.0168 - {0.1110i}} \\{{- 0.1404} - {0.1237i}} & {0.9221 - {0.2312i}} & {{- 0.0433} + {0.2025i}} & {{- 0.1357} + {0.0004i}} \\{{- 0.2862} - {0.1080i}} & {0.1126 + {0.1738i}} & {0.8308 - {0.1386i}} & {0.3917 + {0.02681i}} \\{0.0904 - {0.0666i}} & {0.0957 - {0.0962i}} & {{- 0.3917} - {0.0268i}} & {0.8670 + {0.2513i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{15} = \begin{matrix}{0.9202 - {0.0539i}} & {0.1248 - {0.1626i}} & {0.1330 - {0.1267i}} & {0.2648 - {0.0661i}} \\{{- 0.0267} - {0.2032i}} & {0.9201 + {0.1347i}} & {{- 0.0686} + {0.2650i}} & {{- 0.1340} + {0.0164i}} \\{{- 0.0662} - {0.1713i}} & {0.0380 + {0.2711i}} & {0.8549 - {0.2832i}} & {{- 0.1632} + {0.2315i}} \\{{- 0.2700} - {0.0403i}} & {0.0664 + {0.1175i}} & {0.2315 + {0.1632i}} & {0.8558 - {0.3076i}}\end{matrix}$

${\overset{\sim}{\Theta}}_{16} = \begin{matrix}{1.0000 + {0.0000i}} & {{- 0.0000} - {0.0000i}} & {{- 0.0000} - {0.0000i}} & {0.0000 - {0.0000i}} \\{0.0000 - {0.0000i}} & {0.9250 + {0.2634i}} & {0.0000 + {0.0000i}} & {0.0750 - {0.2634i}} \\{0.0000 - {0.0000i}} & {{- 0.0000} + {0.0000i}} & {1.0000 + {0.0000i}} & {{- 0.0000} + {0.0000i}} \\{0.0000 - {0.0000i}} & {0.0750 - {0.2634i}} & {0.0000 + {0.0000i}} & {0.9250 + {0.2634i}}\end{matrix}$

(6) Time correlation coefficient=0.9

${\overset{\sim}{\Theta}}_{1} = \begin{matrix}{0.9567 + {0.0417i}} & {0.0684 + {0.1542i}} & {0.1036 + {0.1320i}} & {0.1613 + {0.0201i}} \\{{- 0.0607} + {0.1574i}} & {0.9322 + {0.0029i}} & {{- 0.0257} - {0.2653i}} & {0.1414 + {0.1076i}} \\{{- 0.0823} + {0.1462i}} & {0.0777 - {0.2549i}} & {0.9183 + {0.2037i}} & {{- 0.0831} - {0.0956i}} \\{{- 0.1567} + {0.0431i}} & {{- 0.1535} + {0.0894i}} & {0.0956 - {0.0831i}} & {0.9337 - {0.2327i}}\end{matrix}$ ${{\overset{\sim}{\Theta}}_{2} = \begin{matrix}{0.9396 + {0.1527i}} & {0.1496 + {0.1411i}} & {0.2002 + {0.0990i}} & {0.0269 + {0.0316i}} \\{{- 0.1411} + {0.1496i}} & {0.9601 - {0.1555i}} & {{- 0.0529} - {0.0251i}} & {0.0827 - {0.0380i}} \\{{- 0.2116} + {0.0715i}} & {{- 0.0196} - {0.0552i}} & {0.9374 + {0.0301i}} & {0.2570 + {0.0326i}} \\{0.0414 + {0.0033i}} & {{- 0.0853} - {0.0316i}} & {{- 0.2570} - {0.0326i}} & {0.9309 + {0.2374i}}\end{matrix}}\;$ ${\overset{\sim}{\Theta}}_{3} = \begin{matrix}{0.9270 + {0.2411i}} & {0.1542 + {0.0992i}} & {0.0476 + {0.0346i}} & {0.1618 + {0.1392i}} \\{{- 0.1791} + {0.0389i}} & {0.9221 - {0.1792i}} & {0.1375 + {0.1618i}} & {{- 0.0412} + {0.1927i}} \\{{- 0.0308} - {0.0502i}} & {{- 0.0969} + {0.1890i}} & {0.9140 - {0.2683i}} & {{- 0.1277} + {0.1667i}} \\{{- 0.1905} + {0.0962i}} & {0.0356 + {0.1938i}} & {0.1667 + {0.1277i}} & {0.9076 - {0.2188i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{4} = \begin{matrix}{0.9604 + {0.1704i}} & {{- 0.0114} + {0.0036i}} & {{- 0.1093} + {0.0930i}} & {{- 0.1359} + {0.0969i}} \\{0.0114 - {0.0036i}} & {0.9839 - {0.0091i}} & {0.0741 + {0.0750i}} & {{- 0.0930} - {0.1093i}} \\{0.0930 + {0.1093i}} & {{- 0.0750} + {0.0741i}} & {0.9839 - {0.0091i}} & {{- 0.0036} - {0.0114i}} \\{0.0969 + {0.1359i}} & {0.1093 - {0.0930i}} & {{- 0.0036} - {0.0114i}} & {0.9604 + {0.1704i}}\end{matrix}$

${\overset{\sim}{\Theta}}_{5} = \begin{matrix}{0.9301 + {0.2281\; i}} & {{- 0.1266} + {0.0364\; i}} & {{- 0.0870} + {0.2160\; i}} & {0.0718 + {0.0782\; i}} \\{0.1153 + {0.0638\; i}} & {0.9717 + {0.1710\; i}} & {{- 0.0708} - {0.0456\; i}} & {{- 0.0410} + {0.0214\; i}} \\{0.0023 + {0.2329\; i}} & {0.0692 - {0.0479\; i}} & {0.9301 + {0.2341\; i}} & {{- 0.1037} - {0.0895\; i}} \\{{- 0.0448} + {0.0963\; i}} & {0.0460 - {0.0041\; i}} & {0.0895 - {0.1037\; i}} & {0.9681 - {0.1749\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{6} = \begin{matrix}{0.9239 + {0.2135\; i}} & {{- 0.1679} + {0.1484\; i}} & {0.0788 + {0.1302\; i}} & {{- 0.1501} + {0.0700\; i}} \\{0.1484 + {0.1679\; i}} & {0.8870 + {0.2301\; i}} & {{- 0.2150} - {0.0813\; i}} & {0.1032 - {0.2157\; i}} \\{{- 0.0364} + {0.1478\; i}} & {0.2095 - {0.0946\; i}} & {0.9337 + {0.0799\; i}} & {0.2139 - {0.0078\; i}} \\{0.1556 + {0.0567\; i}} & {{- 0.0796} - {0.2255\; i}} & {{- 0.2139} + {0.0078\; i}} & {0.9046 + {0.2265\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{7} = \begin{matrix}{0.9319 + {0.1518\; i}} & {{- 0.1110} + {0.2491\; i}} & {{- 0.1561} - {0.0682\; i}} & {{- 0.0041} + {0.0721\; i}} \\{0.0976 + {0.2546\; i}} & {0.9190 - {0.1100\; i}} & {0.0859 - {0.0053\; i}} & {0.1625 + {0.1876\; i}} \\{0.1227 - {0.1181\; i}} & {{- 0.0773} + {0.0378\; i}} & {0.9244 - {0.2914\; i}} & {{- 0.0789} + {0.1342\; i}} \\{{- 0.0313} + {0.0650\; i}} & {{- 0.1111} + {0.2219\; i}} & {0.1342 + {0.0789\; i}} & {0.9192 - {0.2529\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{8} = \begin{matrix}{0.9222 - {0.0386\; i}} & {0.0068 - {0.0794\; i}} & {{- 0.2353} - {0.0095\; i}} & {{- 0.2918} - {0.0321\; i}} \\{0.0068 - {0.0794\; i}} & {0.9475 - {0.2510\; i}} & {0.1789 - {0.0132\; i}} & {{- 0.0229} + {0.0158\; i}} \\{0.2353 + {0.0095\; i}} & {{- 0.1789} + {0.0132\; i}} & {0.9538 + {0.0073\; i}} & {{- 0.0384} + {0.0335\; i}} \\{0.2918 + {0.0321\; i}} & {0.0229 - {0.0158\; i}} & {{- 0.0384} + {0.0335\; i}} & {0.9286 + {0.2197\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{9} = \begin{matrix}{0.9446 - {0.1610\; i}} & {{- 0.0810} - {0.1798\; i}} & {{- 0.1478} + {0.0594\; i}} & {0.0299 - {0.1293\; i}} \\{0.0699 - {0.1844\; i}} & {0.9625 - {0.0688\; i}} & {0.0910 + {0.0229\; i}} & {{- 0.0815} + {0.1209\; i}} \\{0.1114 + {0.1138\; i}} & {{- 0.0753} - {0.0560\; i}} & {0.9201 + {0.2943\; i}} & {{- 0.1084} - {0.1446\; i}} \\{{- 0.0218} - {0.1309\; i}} & {0.0805 + {0.1215\; i}} & {{0.1446.{- 01084}}\; i} & {0.9396 - {0.2135\; i}}\end{matrix}$

${\overset{\sim}{\Theta}}_{10} = \begin{matrix}{0.9250 - {0.2107\; i}} & {{- 0.1449} - {0.1273\; i}} & {0.0807 - {0.1338\; i}} & {{- 0.1800} - {0.0778\; i}} \\{0.1273 - {0.1449\; i}} & {0.9360 + {0.1128\; i}} & {{- 0.1854} + {0.0814\; i}} & {{- 0.0832} - {0.1614\; i}} \\{{- 0.0375} - {0.1517\; i}} & {0.1886 + {0.0735\; i}} & {0.9310 - {0.0770\; i}} & {0.2485 + {0.0158\; i}} \\{0.1823 - {0.0723\; i}} & {0.0553 - {0.1730\; i}} & {{- 0.2485} - {0.0158\; i}} & {0.9047 + {0.2194\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{11} = \begin{matrix}{0.9006 - {0.2267\; i}} & {{- 0.1948} - {0.1068\; i}} & {{- 0.0646} - {0.2283\; i}} & {0.0718 - {0.1632\; i}} \\{0.2133 - {0.0622\; i}} & {0.9135 + {0.2858\; i}} & {{- 0.1600} + {0.0722\; i}} & {0.0549 - {0.0275\; i}} \\{0.0277 - {0.2357\; i}} & {0.1279 + {0.1202\; i}} & {0.9104 - {0.2221\; i}} & {{- 0.1069} + {0.1526\; i}} \\{{- 0.1234} - {0.1288\; i}} & {{- 0.0402} + {0.0464\; i}} & {0.1526 + {0.1069\; i}} & {0.9197 - {0.2865\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{12} = \begin{matrix}{0.9550 - {0.2020\; i}} & {0.0700 + {0.0209\; i}} & {{- 0.0886} - {0.0854\; i}} & {{- 0.1024} - {0.1278\; i}} \\{{- 0.0700} - {0.0209\; i}} & {0.9582 - {0.0280\; i}} & {0.1337 - {0.1239\; i}} & {0.1703 - {0.1161\; i}} \\{0.0854 - {0.0886\; i}} & {{- 0.1239} - {0.1337\; i}} & {0.9739 - {0.0005\; i}} & {{- 0.0366} + {0.0425\; i}} \\{0.1278 - {0.1024\; i}} & {{- 0.1161} - {0.1703\; i}} & {{- 0.0366} + {0.0425\; i}} & {0.9197 + {0.2859\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{13} = \begin{matrix}{0.9274 - {0.2418\; i}} & {0.1685 - {0.1020\; i}} & {0.0488 - {0.0343\; i}} & {0.1475 - {0.1320\; i}} \\{{- 0.1913} - {0.0470\; i}} & {0.8936 - {0.2315\; i}} & {0.1478 - {0.1782\; i}} & {\;{0.0584 + {0.2282\; i}}} \\{{- 0.0320} + {0.0504\; i}} & {{- 0.1081} - {0.2047\; i}} & {0.9136 + {0.2671\; i}} & {{- 0.1133} - {0.1548\; i}} \\{{- 0.1784} - {0.0857\; i}} & {{- 0.0333} + {0.2332\; i}} & {0.1548 - {0.1133\; i}} & {0.9054 - {0.2207\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{14} = \begin{matrix}{0.9423 - {0.1401\; i}} & {0.1080 - {0.1272\; i}} & {0.2190 - {0.0917\; i}} & {{- 0.0019} - {0.0903\; i}} \\{{- 0.1272} - {0.1080\; i}} & {0.9538 - {0.1788\; i}} & {{- 0.0429} + {0.1382\; i}} & {{- 0.0972} - {0.0097\; i}} \\{{- 0.2197} - {0.0900\; i}} & {0.0674 + {0.1280\; i}} & {0.9094 - {0.0778\; i}} & {0.2982 + {0.0256\; i}} \\{0.0625 - {0.0652\; i}} & {0.0756 - {0.0619\; i}} & {{- 0.2982} - {0.0256\; i}} & {0.9202 + {0.2141\; i}}\end{matrix}$

${\overset{\sim}{\Theta}}_{15} = \begin{matrix}{0.9469 - {0.0481\; i}} & {0.0945 - {0.1362\; i}} & {0.1041 - {0.1137\; i}} & {0.2171 - {0.0514\; i}} \\{{- 0.0295} - {0.1632\; i}} & {0.9480 + {0.1088\; i}} & {{- 0.0542} + {0.2172\; i}} & {{- 0.1063} + {0.0254\; i}} \\{{- 0.0652} - {0.1397\; i}} & {0.0330 + {0.2214\; i}} & {0.9082 - {0.2293\; i}} & {{- 0.1367} + {0.1734\; i}} \\{{- 0.2203} - {0.0356\; i}} & {0.0641 + {0.0885\; i}} & {0.1734 + {0.1367\; i}} & {0.9103 - {0.2466\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{16} = \begin{matrix}{1.0000 + {0.0000\; i}} & {{- 0.0000} - {0.0000\; i}} & {0.0000 - {0.0000\; i}} & {0.0000 - {0.0000\; i}} \\{0 - {0.0000\; i}} & {0.9500 + {0.2179\; i}} & {0.0000 + {0.0000\; i}} & {0.0500 - {0.2179\; i}} \\{0 - {0.0000\; i}} & {0.0000 + {0.0000\; i}} & {1.0000 + {0.0000\; i}} & {{{- 0.0000} + {0.0000i}}\;} \\{{- 0.0000} - {0.0000\; i}} & {0.0500 - {0.2179\; i}} & {0.0000 + {0.0000\; i}} & {0.9500 + {0.2179\; i}}\end{matrix}$

(7) Time correlation coefficient=0.95

${\overset{\sim}{\Theta}}_{1} = \begin{matrix}{09774 + {0.0378\; i}} & {0.0428 + {0.1040\; i}} & {0.0703 + {0.0960\; i}} & {0.1267 + {0.0174\; i}} \\{{- 0.0433} + {0.1038\; i}} & {0.9715 + {0.0287\; i}} & {{- 0.0195} - {0.1746\; i}} & {0.0851 + {0.0679\; i}} \\{{- 0.0618} + {0.1017\; i}} & {0.0488 - {0.1688\; i}} & {0.9603 + {0.1510\; i}} & {{- 0.0692} - {0.0715\; i}} \\{{- 0.1237} + {0.0325\; i}} & {{- 0.0953} + {0.0526\; i}} & {0.0715 - {0.0692\; i}} & {0.9679 - {0.1580\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{2} = \begin{matrix}{0.9705 + {0.1072\; i}} & {0.1078 + {0.1019\; i}} & {0.1407 + {0.1631\; i}} & {0.0045 + {0.0292\; i}} \\{{- 0.1019} + {0.1078\; i}} & {0.9798 - {0.1081\; i}} & {{- 0.0338} - {0.0309\; i}} & {0.0585 - {0.0282\; i}} \\{{- 0.1441} + {0.0548\; i}} & {{- 0.0021} - {0.0457\; i}} & {0.9713 + {0.0172\; i}} & {0.1737 + {0.0129\; i}} \\{0.0238 + {0.0175\; i}} & {{- 0.0613} - {0.0214\; i}} & {{- 0.1737} - {0.0129\; i}} & {09683 + {0.1642\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{3} = \begin{matrix}{0.9669 + {0.1648\; i}} & {0.1083 + {0.0611\; i}} & {0.0323 + {0.0131\; i}} & {0.1140 + {0.0905\; i}} \\{{- 0.1198} + {0.0334\; i}} & {0.9619 - {0.1319\; i}} & {0.0895 + {0.1140\; i}} & {{- 0.0264} + {0.1418\; i}} \\{{- 0.0248} - {0.0245\; i}} & {{- 0.0710} + {0.1263\; i}} & {0.9624 - {0.1805\; i}} & {{- 0.0895} + {0.1049\; i}} \\{{- 0.1272} + {0.0707\; i}} & {0.0299 + {0.1411\; i}} & {0.1049 + {0.0895\; i}} & {0.9565 - {0.1554\; i}}\end{matrix}$

${\overset{\sim}{\Theta}}_{4} = \begin{matrix}{097940.1331\; i} & {{- 0.0034} + {0.0013\; i}} & {{- 0.0760} + {0.0666\; i}} & {{- 0.0903} + {0.0691\; i}} \\{0.0034 - {0.0013\; i}} & {0.9913 - {0.0028\; i}} & {0.0595 + {0.0596\; i}} & {{- 0.0666} - {0.07601\; i}} \\{0.0666 + {0.0760\; i}} & {{- 0.0596} + {0.0595\; i}} & {0.9913 - {0.0028\; i}} & {{- 0.0013} - {0.0034\; i}} \\{0.0691 + {0.0903\mspace{11mu} i}} & {0.0760 - {0.0666\; i}} & {{- 0.0013} - {0.0034\; i}} & {0.9794 + {0.1331\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{5} = \begin{matrix}{0.9660 + {0.1576\; i}} & {{- 0.0934} + {0.0306\; i}} & {{- 0.0508} + {0.1548\; i}} & {0.0496 + {0.0585\; i}} \\{0.0876 + {0.0444\; i}} & {0.9845 + {0.1253\; i}} & {{- 0.0552} - {0.0370\; i}} & {{- 0.0282} + {0.0130\; i}} \\{0.0123 + {0.1624\; i}} & {0.0553 - {0.0368\; i}} & {0.9656 + {0.1638\; i}} & {{- 0.0742} - {0.0662\; i}} \\{{- 0.0351} + {0.0682\; i}} & {0.0310 - {0.0012\; i}} & {0.0662 - {0.0742\; i}} & {0.9826 - {0.1330\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{6} = \begin{matrix}{0.9658 + {0.1329\; i}} & {{- 0.1178} + {0.1097\; i}} & {0.0472 + {0.0730\; i}} & {{- 0.1166} + {0.0496\; i}} \\{0.1097 + {0.1178\; i}} & {0.9496 + {0.1469\; i}} & {{- 0.1458} - {0.0542\; i}} & {0.0652 - {0.1494\; i}} \\{{- 0.0183} + {0.0850\; i}} & {0.1414 - {0.0648\; i}} & {0.9716 + {0.0302\; i}} & {0.1527 - {0.0038\; i}} \\{0.1175 + {0.0473\; i}} & {{- 0.0596} - {0.1517\; i}} & {{- 0.1527} + {0.0038\; i}} & {0.9533 + {0.1593\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{7} = \begin{matrix}{0.9765 + {0.0860\; i}} & {{- 0.0629} + {0.1474\; i}} & {{- 0.0954} - {0.0494\; i}} & {0.0034 + {0.0423\; i}} \\{0.0597 + {0.1487\; i}} & {0.9703 - {0.0737\; i}} & {0.0476 + {0.0032\; i}} & {0.0995 + {0.1236\; i}} \\{0.0821 - {0.0693\; i}} & {{- 0.0452} + {0.0152\; i}} & {0.9681 - {0.1906\; i}} & {{- 0.0644} + {0.0917\; i}} \\{{- 0.0130} + {0.0404\; i}} & {{- 0.0761} + {0.1392\; i}} & {0.0917 + {0.0644\; i}} & {0.9643 - {0.1749\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{8} = \begin{matrix}{0.9678 - {0.0104\; i}} & {0.0076 - {0.0333\; i}} & {{- 0.1621} - {0.0037\; i}} & {{- 0.1885} - {0.0160\; i}} \\{0.0076 - {0.0333\; i}} & {0.9761 - {0.1656\; i}} & {0.1357 - {0.0085\; i}} & {{- 0.0069} + {0.0046\; i}} \\{0.1621 + {0.0037\; i}} & {{- 0.1357} + {0.0085\; i}} & {0.9770 + {0.0034\; i}} & {{- 0.0169} + {0.0195\; i}} \\{0.1885 + {0.0160\; i}} & {0.0069 - {0.0046\; i}} & {{- 0.0169} + {0.0195\; i}} & {0.9687 + {0.1586\; i}}\end{matrix}$

${\overset{\sim}{\Theta}}_{9} = \begin{matrix}{0.9802 - {0.0878\; i}} & {{- 0.0504} - {0.1161\; i}} & {{- 0.0931} + {0.0463\; i}} & {0.0188 - {0.0664\; i}} \\{0.0464 - {0.1177\; i}} & {0.9827 - {0.0501\; i}} & {0.0540 + {0.0074\; i}} & {{- 0.0596} + {0.0958\; i}} \\{0.0784 + {0.0683\; i}} & {{- 0.0472} - {0.0272\; i}} & {0.9666 + {0.1925\; i}} & {{- 0.0792} - {0.0927\; i}} \\{{- 0.0080} - {0.0685\; i}} & {0.0657 + {0.0917\; i}} & {0.0927 - {0.0792\; i}} & {0.9726 - {0.1473\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{10} = \begin{matrix}{0.9661 - {0.1316\; i}} & {{- 0.1066} - {0.0970\; i}} & {0.0481 - {0.0743\; i}} & {{- 0.1330} - {0.0548\; i}} \\{0.0970 - {0.1066\; i}} & {0.9664 + {0.0872\; i}} & {{- 0.1309} + {0.0552\; i}} & {{- 0.0566} - {0.1195\; i}} \\{{- 0.0186} - {0.0865\; i}} & {0.1316 + {0.0535\; i}} & {0.9706 - {0.0291\; i}} & {0.1700 + {0.0081\; i}} \\{0.1328 - {0.0553\; i}} & {0.0445 - {0.1246\; i}} & {{- 0.1700} - {0.0081\; i}} & {0.9530 + {0.1571\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{11} = \begin{matrix}{0.9567 - {0.1574\; i}} & {{- 0.1273} - {0.0627\; i}} & {{- 0.0433} - {0.1588\; i}} & {0.0524 - {0.1000\; i}} \\{0.1343 - {0.1456\; i}} & {0.9657 + {0.1828\; i}} & {{- 0.0985} + {0.0525\; i}} & {0.0357 - {0.0086i}} \\{0.0207 - {0.1633\; i}} & {0.0862 + {0.0709\; i}} & {0.9588 - {0.1596\; i}} & {{- 0.0775} + {0.0984\; i}} \\{{- 0.0723} - {0.0867\; i}} & {{- 0.0297} + {0.0216\; i}} & {0.0984 + {0.0775\; i}} & {0.9664 - {0.1907\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{12} = \begin{matrix}{0.9781 - {0.1416\; i}} & {0.0301 + {0.0113\; i}} & {{- 0.0701} - {0.649\; i}} & {{- 0.0764} - {0.0849\; i}} \\{{- 0.0301} - {0.0113\; i}} & {0.9833 - {0.0066\; i}} & {0.0864 - {0.0825\; i}} & {0.1049 - {0.0828\; i}} \\{0.0649 - {0.0701\; i}} & {{- 0.0825} - {0.0864\; i}} & {0.9878 + {0.0009\; i}} & {{- 0.0159} - {{+ 0.0226}\; i}} \\{0.0849 - {0.0764\; i}} & {{- 0.0828} - {0.1049\; i}} & {{- 0.0159} + {0.0226\; i}} & {0.9658 + {0.1886\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{13} = \begin{matrix}{0.9672 - {0.1650\; i}} & {0.1156 - {0.0615\; i}} & {0.0328 - {0.0130\; i}} & {0.1064 - {0.0876\; i}} \\{{- 0.1252} - {0.0383\; i}} & {0.9521 - {0.1597\; i}} & {0.0944 - {0.1225\; i}} & {0.0392 + {0.1593\; i}} \\{{- 0.0253} + {0.0246\; i}} & {{- 0.0771} - {0.1341\; i}} & {0.9622 + {0.1801\; i}} & {{- 0.0819} - {0.0994\; i}} \\{{- 0.1217} - {0.0648\; i}} & {{- 0.0247} + {0.1622\; i}} & {0.0994 - {0.0819\; i}} & {0.9552 - {0.1582\; i}}\end{matrix}$

${\overset{\sim}{\Theta}}_{14} = \begin{matrix}{0.9714 - {0.1011\; i}} & {0.0849 - {0.0985\; i}} & {0.1470 - {0.0596\; i}} & {{- 0.0115} - {0.0628\; i}} \\{{- 0.0985} - {0.0849\; i}} & {0.9786 - {0.1196\; i}} & {{- 0.0308} + {0.0794\; i}} & {{- 0.0607} - {0.0128\; i}} \\{{- 0.1461} - {0.0619\; i}} & {0.0344 + {0.0779\; i}} & {0.9640 - {0.0304\; i}} & {0.1928 + {0.0150\; i}} \\{0.0363 - {0.0525\; i}} & {0.0520 - {0.0338\; i}} & {{- 0.1928} - {0.0150\; i}} & {0.9645 + {0.1561\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{15} = \begin{matrix}{0.9733 - {0.0393\; i}} & {0.0598 - {0.0987\; i}} & {0.0711 - {0.0896\; i}} & {0.1534 - {0.0344\; i}} \\{{- 0.0275} - {0.1121\; i}} & {0.9748 + {0.0743\; i}} & {{- 0.0361} + {0.1534\; i}} & {{- 0.0732} + {0.0282\; i}} \\{{- 0.0555} - {0.1000\; i}} & {0.0254 + {0.1556\; i}} & {0.9569 - {0.1597\; i}} & {{- 0.0955} + {0.1083\; i}} \\{{- 0.1549} - {0.0269\; i}} & {0.0541 + {0.0569\; i}} & {0.1083 + {0.0955\; i}} & {0.9590 - {0.1688\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{16} = \begin{matrix}{1.0000 + {0.0000\; i}} & {{- 0.0000} - {0.0000\; i}} & {0.0000 - {0.0000\; i}} & {0.0000 - {0.0000\; i}} \\{0.0000 - {0.0000\; i}} & {0.9750 + {0.1561\; i}} & {{- 0.0000} + {0.0000\; i}} & {0.0250 - {0.1561\; i}} \\{{- 0.0000} - {0.0000\; i}} & {0.0000 + {0.0000\; i}} & {1.0000 - {0.0000\; i}} & {{- 0.0000} + {0.0000\; i}} \\{{- 0.0000} - {0.0000\; i}} & {0.0250 - {0.1561\; i}} & {0.0000 + {0.0000\; i}} & {0.9750 + {0.1561\; i}}\end{matrix}$

2) Where the number of feedback bits B=4, {θ}={Θ₁, . . . , Θ₂ _(B) }includes diagonal unitary matrices, and the second scheme, that is, theabove Equation 16 is used, examples of the codebook {{tilde over(θ)}}={{tilde over (Θ)}₁, . . . , {tilde over (Θ)}₂ _(B) } that isupdated according to the time correlation coefficient may follow as:

(1) Time correlation coefficient=0

${\overset{\sim}{\Theta}}_{1} = \begin{matrix}{0.9239 + {0.3827\; i}} & 0 & 0 & 0 \\0 & {0.3827 + {0.9239\; i}} & 0 & 0 \\0 & 0 & {0.0000 + {1.0000\; i}} & 0 \\0 & 0 & 0 & {{- 1.0000} + {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{2} = \begin{matrix}{0.7071 + {0.7071\; i}} & 0 & 0 & 0 \\0 & {{- 0.7071} + {0.7071\; i}} & 0 & 0 \\0 & 0 & {{- 1.0000} + {0.0000\; i}} & 0 \\0 & 0 & 0 & {1.0000 + {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{3} = \begin{matrix}{0.3827 + {0.9239\; i}} & 0 & 0 & 0 \\0 & {{- 0.9239} - {0.3827\; i}} & 0 & 0 \\0 & 0 & {{- 0.0000} - {1.0000\; i}} & 0 \\0 & 0 & 0 & {{- 1.0000} + {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{4 =}\begin{matrix}{{- 0.0000} + {1.0000\; i}} & 0 & 0 & 0 \\0 & {{- 0.0000} - {1.0000\; i}} & 0 & 0 \\0 & 0 & {1.0000 - {0.0000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{5} = \begin{matrix}{{- 0.3827} + {0.9239\; i}} & 0 & 0 & 0 \\0 & {0.9239 - {0.3827\; i}} & 0 & 0 \\0 & 0 & {0.0000 + {1.0000\; i}} & 0 \\0 & 0 & 0 & {{- 1.0000} + {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{6} = \begin{matrix}{{- 0.7071} + {0.7071\; i}} & 0 & 0 & 0 \\0 & {0.7071 + {0.7071\; i}} & 0 & 0 \\0 & 0 & {{- 1.0000} + {0.0000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$

${\overset{\sim}{\Theta}}_{7} = \begin{matrix}{{- 0.9239} + {0.3827\; i}} & 0 & 0 & 0 \\0 & {{- 0.3827} + {0.9239\; i}} & 0 & 0 \\0 & 0 & {{- 0.0000} - {1.0000\; i}} & 0 \\0 & 0 & 0 & {{- 1.0000} + {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{8} = \begin{matrix}{{- 1.0000} - {0.0000\; i}} & 0 & 0 & 0 \\0 & {{- 1.0000} + {0.0000\; i}} & 0 & 0 \\0 & 0 & {1.0000 - {0.0000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{9} = \begin{matrix}{{- 0.9239} - {0.3827\; i}} & 0 & 0 & 0 \\0 & {{- 0.3827} - {0.9239\; i}} & 0 & 0 \\0 & 0 & {0.0000 + {1.0000\; i}} & 0 \\0 & 0 & 0 & {{- 1.0000} + {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{10} = \begin{matrix}{{- 0.7071} - {0.7071\; i}} & 0 & 0 & 0 \\0 & {0.7071 - {0.7071\; i}} & 0 & 0 \\0 & 0 & {{- 1.0000} + {0.0000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{11} = \begin{matrix}{{- 0.3827} - {0.9239\; i}} & 0 & 0 & 0 \\0 & {0.9239 + {0.3827\; i}} & 0 & 0 \\0 & 0 & {{- 0.0000} - {1.0000\; i}} & 0 \\0 & 0 & 0 & {{- 1.0000} + {0.0000\; i}}\end{matrix}$

${\overset{\sim}{\Theta}}_{12} = \begin{matrix}{0.0000 - {1.0000\; i}} & 0 & 0 & 0 \\0 & {0.0000 + {1.0000\; i}} & 0 & 0 \\0 & 0 & {1.0000 - {0.0000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{13} = \begin{matrix}{0.3827 - {0.9239\; i}} & 0 & 0 & 0 \\0 & {{- 0.9239} + {0.3827\; i}} & 0 & 0 \\0 & 0 & {0.0000 + {1.0000\; i}} & 0 \\0 & 0 & 0 & {{- 1.0000} + {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{14} = \begin{matrix}{0.7071 - {0.7071\; i}} & 0 & 0 & 0 \\0 & {{- 0.7071} - {0.7071\; i}} & 0 & 0 \\0 & 0 & {{- 1.0000} + {0.0000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{15} = \begin{matrix}{0.9239 - {0.3827\; i}} & 0 & 0 & 0 \\0 & {0.3827 - {0.9239\; i}} & 0 & 0 \\0 & 0 & {{- 0.0000} - {1.0000\; i}} & 0 \\0 & 0 & 0 & {{- 1.0000} + {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{16} = \begin{matrix}{1.0000 + {0.0000\; i}} & 0 & 0 & 0 \\0 & {1.0000 - {0.0000\; i}} & 0 & 0 \\0 & 0 & {1.0000 - {0.0000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$

(2) Time correlation coefficient=0.7

${\overset{\sim}{\Theta}}_{1} = \begin{matrix}{0.9804 + {0.1970\; i}} & 0 & 0 & 0 \\0 & {0.8277 + {0.5611\; i}} & 0 & 0 \\0 & 0 & {0.7000 + {0.7141\; i}} & 0 \\0 & 0 & 0 & {{- 1.0000} + {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{2} = \begin{matrix}{0.9223 + {0.3865\; i}} & 0 & 0 & 0 \\0 & {0.3603 + {0.9328\; i}} & 0 & 0 \\0 & 0 & {{- 1.0000} + {0.0000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{3} = \begin{matrix}{0.8277 + {0.5611\; i}} & 0 & 0 & 0 \\0 & {0.1456 - {0.9893\; i}} & 0 & 0 \\0 & 0 & {0.7000 - {0.7141\; i}} & 0 \\0 & 0 & 0 & {{- 1.0000} + {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{4} = \begin{matrix}{0.7000 + {0.7141\; i}} & 0 & 0 & 0 \\0 & {0.7000 - {0.7141\; i}} & 0 & 0 \\0 & 0 & {1.0000 - {0.0000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{5} = \begin{matrix}{0.5431 + {0.8397\; i}} & 0 & 0 & 0 \\0 & {0.9804 - {0.1970\; i}} & 0 & 0 \\0 & 0 & {0.7000 + {0.7141\; i}} & 0 \\0 & 0 & 0 & {{- 1.0000} + {0.0000\; i}}\end{matrix}$

${\overset{\sim}{\Theta}}_{6} = \begin{matrix}{0.3603 + {0.9328\; i}} & 0 & 0 & 0 \\0 & {0.9223 + {0.3865\; i}} & 0 & 0 \\0 & 0 & {{- 1.0000} + {0.000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{7} = \begin{matrix}{0.1456 + {0.9893\; i}} & 0 & 0 & 0 \\0 & {0.5431 + {0.8397\; i}} & 0 & 0 \\0 & 0 & {0.7000 - {0.7141\; i}} & 0 \\0 & 0 & 0 & {{- 1.0000} + {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{8} = \begin{matrix}{{- 1.0000} - {0.0000\; i}} & 0 & 0 & 0 \\0 & {{- 1.0000} + {0.0000\; i}} & 0 & 0 \\0 & 0 & {1.0000 - {0.0000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{9} = \begin{matrix}{0.1456 - {0.9893\; i}} & 0 & 0 & 0 \\0 & {0.5431 - {0.8397\; i}} & 0 & 0 \\0 & 0 & {0.7000 + {{.0}{.7141}\; i}} & 0 \\0 & 0 & 0 & {{- 1.0000} + {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{10} = \begin{matrix}{0.3603 - {0.9328\; i}} & 0 & 0 & 0 \\0 & {0.9223 - {0.3865\; i}} & 0 & 0 \\0 & 0 & {{- 1.0000} + {0.0000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$

${\overset{\sim}{\Theta}}_{11} = \begin{matrix}{0.5431 - {0.8397\; i}} & 0 & 0 & 0 \\0 & {0.9804 + {0.1970\; i}} & 0 & 0 \\0 & 0 & {0.7000 - {0.7141\; i}} & 0 \\0 & 0 & 0 & {{- 1.000} + {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{12} = \begin{matrix}{0.7000 - {0.7141\; i}} & 0 & 0 & 0 \\0 & {0.7000 + {0.7141\; i}} & 0 & 0 \\0 & 0 & {1.0000 - {0.0000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{13} = \begin{matrix}{0.8277 - {0.5611\; i}} & 0 & 0 & 0 \\0 & {0.1456 + {0.9893\; i}} & 0 & 0 \\0 & 0 & {0.7000 + {0.7141\; i}} & 0 \\0 & 0 & 0 & {{- 1.0000} + {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{14} = \begin{matrix}{0.9223 - {0.3865\; i}} & 0 & 0 & 0 \\0 & {0.3603 - {0.9328\; i}} & 0 & 0 \\0 & 0 & {{- 1.0000} + {0.0000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{15} = \begin{matrix}{0.9804 - {0.1970\; i}} & 0 & 0 & 0 \\0 & {0.8277 - {0.5611\; i}} & 0 & 0 \\0 & 0 & {0.7000 - {0.7141\; i}} & 0 \\0 & 0 & 0 & {{- 1.0000} + {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{16} = \begin{matrix}{1.0000 + {0.0000\; i}} & 0 & 0 & 0 \\0 & {1.0000 - {0.0000\; i}} & 0 & 0 \\0 & 0 & {1.0000 - {0.0000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$

(3) Time correlation coefficient=0.75

${\overset{\sim}{\Theta}}_{1} = \begin{matrix}{0.9831 + {0.1828\; i}} & 0 & 0 & 0 \\0 & {0.8540 + {0.5203\; i}} & 0 & 0 \\0 & 0 & {0.7500 + {0.6614\; i}} & 0 \\0 & 0 & 0 & {1.0000 + {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{2} = \begin{matrix}{0.9335 + {0.3586\; i}} & 0 & 0 & 0 \\0 & {0.5167 + {0.8561\; i}} & 0 & 0 \\0 & 0 & {1.0000 + {0.0000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{3} = \begin{matrix}{0.8540 + {0.5203\; i}} & 0 & 0 & 0 \\0 & {0.4811 - {0.8767\; i}} & 0 & 0 \\0 & 0 & {0.7500 - {0.6614\; i}} & 0 \\0 & 0 & 0 & {1.0000 + {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{4} = \begin{matrix}{0.7500 + {0.6614\; i}} & 0 & 0 & 0 \\0 & {0.7500 - {0.6614\; i}} & 0 & 0 \\0 & 0 & {1.0000 - {0.0000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$

${\overset{\sim}{\Theta}}_{5} = \begin{matrix}{0.6309 + {0.7759\; i}} & 0 & 0 & 0 \\0 & {0.9831 - {0.1828\; i}} & 0 & 0 \\0 & 0 & {0.7500 + {0.6614\; i}} & 0 \\0 & 0 & 0 & {1.0000 + {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{6} = \begin{matrix}{0.5167 + {0.8561\; i}} & 0 & 0 & 0 \\0 & {0.9335 + {0.3586\; i}} & 0 & 0 \\0 & 0 & {1.0000 + {0.0000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{7} = \begin{matrix}{0.4811 + {0.8767\; i}} & 0 & 0 & 0 \\0 & {0.6309 + {0.7759\; i}} & 0 & 0 \\0 & 0 & {0.7500 - {0.6614\; i}} & 0 \\0 & 0 & 0 & {1.0000 + {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{8} = \begin{matrix}{1.0000 - {0.0000\; i}} & 0 & 0 & 0 \\0 & {1.0000 + {0.0000\; i}} & 0 & 0 \\0 & 0 & {1.0000 - {0.0000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{9} = \begin{matrix}{0.4811 - {0.8767\; i}} & 0 & 0 & 0 \\0 & {0.6309 - {0.7759\; i}} & 0 & 0 \\0 & 0 & {0.7500 + {0.6614\; i}} & 0 \\0 & 0 & 0 & {1.0000 + {0.0000\; i}}\end{matrix}$

${\overset{\sim}{\Theta}}_{10} = \begin{matrix}{0.5167 - {0.8561i}} & 0 & 0 & 0 \\0 & {0.9335 - {0.3586\; i}} & 0 & 0 \\0 & 0 & {1.0000 + {0.0000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{11} = \begin{matrix}{0.6309 - {0.7759\; i}} & 0 & 0 & 0 \\0 & {0.9831 + {0.1828\; i}} & 0 & 0 \\0 & 0 & {0.7500 - {0.6614\; i}} & 0 \\0 & 0 & 0 & {1.0000 + {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{12} = \begin{matrix}{0.7500 - {0.6614\; i}} & 0 & 0 & 0 \\0 & {0.7500 + {0.6614\; i}} & 0 & 0 \\0 & 0 & {1.000 - {0.0000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{13} = \begin{matrix}{0.8540 - {0.5203\; i}} & 0 & 0 & 0 \\0 & {0.4811 + {0.8767\; i}} & 0 & 0 \\0 & 0 & {0.7500 + {0.6614\; i}} & 0 \\0 & 0 & 0 & {1.000 + {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{14} = \begin{matrix}{0.9335 - {0.3586\; i}} & 0 & 0 & 0 \\0 & {0.5167 - {0.8561\; i}} & 0 & 0 \\0 & 0 & {1.0000 + {0.0000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$

${\overset{\sim}{\Theta}}_{15} = \begin{matrix}{0.9831 - {0.1828\; i}} & 0 & 0 & 0 \\0 & {0.8540 - {0.5203\; i}} & 0 & 0 \\0 & 0 & {0.7500 - {0.6614\; i}} & 0 \\0 & 0 & 0 & {1.0000 + {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{16} = \begin{matrix}{1.0000 + {0.0000\; i}} & 0 & 0 & 0 \\0 & {1.0000 - {0.0000\; i}} & 0 & 0 \\0 & 0 & {1.0000 - {0.0000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$

(4) Time correlation coefficient=0.8

${\overset{\sim}{\Theta}}_{1} = \begin{matrix}{0.9859 + {0.1672\; i}} & 0 & 0 & 0 \\0 & {0.8805 + {0.4740\; i}} & 0 & 0 \\0 & 0 & {0.8000 + {0.6000\; i}} & 0 \\0 & 0 & 0 & {1.0000 + {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{2} = \begin{matrix}{0.9449 + {0.3274\; i}} & 0 & 0 & 0 \\0 & {0.6630 + {0.7486\; i}} & 0 & 0 \\0 & 0 & {1.0000 + {0.0000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{3} = \begin{matrix}{0.8805 + {0.4740\; i}} & 0 & 0 & 0 \\0 & {0.7306 - {0.6828\; i}} & 0 & 0 \\0 & 0 & {0.8000 - {0.6000\; i}} & 0 \\0 & 0 & 0 & {1.0000 + {0.0000\; i}}\end{matrix}$

${\overset{\sim}{\Theta}}_{4} = \begin{matrix}{0.8000 + {0.6000\; i}} & 0 & 0 & 0 \\0 & {0.8000 - {0.6000\; i}} & 0 & 0 \\0 & 0 & {1.0000 - {0.0000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{5} = \begin{matrix}{0.7171 + {0.6969\; i}} & 0 & 0 & 0 \\0 & {0.9859 - {0.1672\; i}} & 0 & 0 \\0 & 0 & {0.8000 + {0.6000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{6} = \begin{matrix}{0.6630 + {0.7486\; i}} & 0 & 0 & 0 \\0 & {0.9449 + {0.3274\; i}} & 0 & 0 \\0 & 0 & {1.0000 + {0.0000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{7} = \begin{matrix}{0.7306 + {0.6828\; i}} & 0 & 0 & 0 \\0 & {0.7171 + {0.6969\; i}} & 0 & 0 \\0 & 0 & {0.8000 - {0.6000\; i}} & 0 \\0 & 0 & 0 & {1.0000 + {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{8} = \begin{matrix}{1.0000 - {0.0000\; i}} & 0 & 0 & 0 \\0 & {1.0000 + {0.0000\; i}} & 0 & 0 \\0 & 0 & {1.0000 - {0.0000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$

${\overset{\sim}{\Theta}}_{9} = \begin{matrix}{0.7306 - {0.6828\; i}} & 0 & 0 & 0 \\0 & {0.7171 - {0.6969\; i}} & 0 & 0 \\0 & 0 & {0.8000 + {0.6000\; i}} & 0 \\0 & 0 & 0 & {1.0000 + {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{10} = \begin{matrix}{0.6630 - {0.7486\; i}} & 0 & 0 & 0 \\0 & {0.9449 - {0.3274\; i}} & 0 & 0 \\0 & 0 & {1.0000 + {0.0000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{11} = \begin{matrix}{0.7171 - {0.6969\; i}} & 0 & 0 & 0 \\0 & {0.9859 + {0.1672\; i}} & 0 & 0 \\0 & 0 & {0.8000 - {0.6000\; i}} & 0 \\0 & 0 & 0 & {1.0000 + {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{12} = \begin{matrix}{0.8000 - {0.6000\; i}} & 0 & 0 & 0 \\0 & {0.8000 + {0.6000\; i}} & 0 & 0 \\0 & 0 & {1.0000 - {0.0000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{13} = \begin{matrix}{0.8805 - {0.4740\; i}} & 0 & 0 & 0 \\0 & {0.7306 + {0.6828\; i}} & 0 & 0 \\0 & 0 & {0.8000 + {0.6000\; i}} & 0 \\0 & 0 & 0 & {1.0000 + {0.0000\; i}}\end{matrix}$

${\overset{\sim}{\Theta}}_{14} = \begin{matrix}{0.9449 - {0.3274\; i}} & 0 & 0 & 0 \\0 & {0.6630 - {0.7486\; i}} & 0 & 0 \\0 & 0 & {1.0000 + {0.0000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{15} = \begin{matrix}{0.9859 - {0.1672\; i}} & 0 & 0 & 0 \\0 & {0.8805 - {0.4740\; i}} & 0 & 0 \\0 & 0 & {0.8000 - {0.6000\; i}} & 0 \\0 & 0 & 0 & {1.0000 + {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{16} = \begin{matrix}{1.0000 + {0.0000\; i}} & 0 & 0 & 0 \\0 & {1.0000 - {0.0000\; i}} & 0 & 0 \\0 & 0 & {1.0000 - {0.0000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$

(5) Time correlation coefficient=0.85

${\overset{\sim}{\Theta}}_{1} = \begin{matrix}{0.9888 + {0.1491\; i}} & 0 & 0 & 0 \\0 & {0.9075 + {0.4200\; i}} & 0 & 0 \\0 & 0 & {0.8500 + {0.5268\; i}} & 0 \\0 & 0 & 0 & {1.0000 + {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{2} = \begin{matrix}{0.9566 + {0.2915i}} & 0 & 0 & 0 \\0 & {0.7885 + {0.6151\; i}} & 0 & 0 \\0 & 0 & {1.0000 + {0.0000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$

${\overset{\sim}{\Theta}}_{3} = \begin{matrix}{0.9075 + {0.4200\; i}} & 0 & 0 & 0 \\0 & {0.8744 - {0.4852\; i}} & 0 & 0 \\0 & 0 & {0.8500 - {0.5268\; i}} & 0 \\0 & 0 & 0 & {1.0000 + {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{4} = \begin{matrix}{0.8500 + {0.5268\; i}} & 0 & 0 & 0 \\0 & {0.8500 - {0.5268\; i}} & 0 & 0 \\0 & 0 & {1.0000 - {0.0000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{5} = \begin{matrix}{0.7998 + {0.6003\; i}} & 0 & 0 & 0 \\0 & {0.9888 - {0.1491\; i}} & 0 & 0 \\0 & 0 & {0.8500 + {0.5268\; i}} & 0 \\0 & 0 & 0 & {1.0000 + {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{6} = \begin{matrix}{0.7885 + {0.6151\; i}} & 0 & 0 & 0 \\0 & {0.9566 + {{.02915}\; i}} & 0 & 0 \\0 & 0 & {1.0000 + {0.0000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{7} = \begin{matrix}{0.8744 + {0.4852\; i}} & 0 & 0 & 0 \\0 & {0.7998 + {0.6003\; i}} & 0 & 0 \\0 & 0 & {0.8500 - {0.5268\; i}} & 0 \\0 & 0 & 0 & {1.0000 + {0.0000\; i}}\end{matrix}$

${\overset{\sim}{\Theta}}_{8} = \begin{matrix}{1.0000 - {0.0000\; i}} & 0 & 0 & 0 \\0 & {1.0000 + {0.0000\; i}} & 0 & 0 \\0 & 0 & {1.0000 - {0.0000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{9} = \begin{matrix}{0.8744 - {0.4852\; i}} & 0 & 0 & 0 \\0 & {0.7998 - {0.6003\; i}} & 0 & 0 \\0 & 0 & {0.8500 + {0.5268\; i}} & 0 \\0 & 0 & 0 & {1.0000 + {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{10} = \begin{matrix}{0.7885 - {0.6151\; i}} & 0 & 0 & 0 \\0 & {0.9566 - {0.29151\; i}} & 0 & 0 \\0 & 0 & {1.0000 + {0.0000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{11} = \begin{matrix}{0.7998 - {0.6003\; i}} & 0 & 0 & 0 \\0 & {0.9888 + {0.1491\; i}} & 0 & 0 \\0 & 0 & {0.8500 - {0.5268\; i}} & 0 \\0 & 0 & 0 & {1.0000 + {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{12} = \begin{matrix}{0.8500 - {0.5268\; i}} & 0 & 0 & 0 \\0 & {0.8500 + {0.5268\; i}} & 0 & 0 \\0 & 0 & {1.0000 - {0.0000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$

${\overset{\sim}{\Theta}}_{13} = \begin{matrix}{0.9075 - {0.4200\; i}} & 0 & 0 & 0 \\0 & {0.8744 + {0.4852\; i}} & 0 & 0 \\0 & 0 & {0.8500 + {0.5268\; i}} & 0 \\0 & 0 & 0 & {1.0000 + {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{14} = \begin{matrix}{0.9566 - {0.2915\; i}} & 0 & 0 & 0 \\0 & {0.7885 - {0.6151\; i}} & 0 & 0 \\0 & 0 & {1.0000 + {0.0000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{15} = \begin{matrix}{0.9888 - {0.1491\; i}} & 0 & 0 & 0 \\0 & {0.9075 - {0.4200\; i}} & 0 & 0 \\0 & 0 & {0.8500 - {0.5268\; i}} & 0 \\0 & 0 & 0 & {1.0000 + {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{16} = \begin{matrix}{1.0000 + {0.0000\; i}} & 0 & 0 & 0 \\0 & {1.0000 - {0.0000\; i}} & 0 & 0 \\0 & 0 & {1.0000 - {0.0000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$

(6) Time correlation coefficient=0.9

${\overset{\sim}{\Theta}}_{1} = \begin{matrix}{0.9919 + {0.1270\; i}} & 0 & 0 & 0 \\0 & {0.9356 + {0.3532\; i}} & 0 & 0 \\0 & 0 & {0.9000 + {0.4359i}} & 0 \\0 & 0 & 0 & {1.0000 + {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{2} = \begin{matrix}{0.9690 + {0.2472\; i}} & 0 & 0 & 0 \\0 & {0.8869 + {0.4619\; i}} & 0 & 0 \\0 & 0 & {1.0000 + {0.0000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{3} = \begin{matrix}{0.9356 + {0.3532i}} & 0 & 0 & 0 \\0 & {0.9481 - {0.3180\; i}} & 0 & 0 \\0 & 0 & {0.9000 - {0.4359\; i}} & 0 \\0 & 0 & 0 & {1.0000 + {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{4} = \begin{matrix}{0.9000 + {0.4359\; i}} & 0 & 0 & 0 \\0 & {0.9000 - {0.4359\; i}} & 0 & 0 \\0 & 0 & {1.0000 - {0.0000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{5} = \begin{matrix}{0.8765 + {0.4814\; i}} & 0 & 0 & 0 \\0 & {0.9919 - {0.1270\; i}} & 0 & 0 \\0 & 0 & {0.9000 + {0.4359\; i}} & 0 \\0 & 0 & 0 & {1.0000 + {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{6} = \begin{matrix}{0.8869 + {0.4619i}} & 0 & 0 & 0 \\0 & {0.9690 + {{.02472}\; i}} & 0 & 0 \\0 & 0 & {1.0000 + {0.0000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$

${\overset{\sim}{\Theta}}_{7} = \begin{matrix}{0.9481 + {0.3180\; i}} & 0 & 0 & 0 \\0 & {0.8765 + {0.4814\; i}} & 0 & 0 \\0 & 0 & {0.9000 - {0.4359\; i}} & 0 \\0 & 0 & 0 & {1.0000 + {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{8} = \begin{matrix}{1.0000 - {0.0000\; i}} & 0 & 0 & 0 \\0 & {1.0000 + {0.0000\; i}} & 0 & 0 \\0 & 0 & {1.0000 - {0.0000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{9} = \begin{matrix}{0.9481 - {0.3180\; i}} & 0 & 0 & 0 \\0 & {0.8765 - {0.4814i}} & 0 & 0 \\0 & 0 & {0.9000 + {0.4359\; i}} & 0 \\0 & 0 & 0 & {1.0000 + {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{10} = \begin{matrix}{0.8869 - {0.4619\; i}} & 0 & 0 & 0 \\0 & {0.9690 - {0.2472\; i}} & 0 & 0 \\0 & 0 & {1.0000 + {0.0000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{11} = \begin{matrix}{0.8765 - {0.4814\; i}} & 0 & 0 & 0 \\0 & {0.9919 + {0.1270\; i}} & 0 & 0 \\0 & 0 & {0.9000 - {0.4359\; i}} & 0 \\0 & 0 & 0 & {1.0000 + {0.0000\; i}}\end{matrix}$

${\overset{\sim}{\Theta}}_{12} = \begin{matrix}{0.9000 - {0.4359\; i}} & 0 & 0 & 0 \\0 & {0.9000 + {0.4359\; i}} & 0 & 0 \\0 & 0 & {1.0000 - {0.0000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{13} = \begin{matrix}{0.9356 - {0.3532i}} & 0 & 0 & 0 \\0 & {0.9481 + {0.3180\; i}} & 0 & 0 \\0 & 0 & {0.9000 + {0.4359\; i}} & 0 \\0 & 0 & 0 & {1.0000 + {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{14} = \begin{matrix}{0.9690 - {0.2472\; i}} & 0 & 0 & 0 \\0 & {0.8869 - {0.4619\; i}} & 0 & 0 \\0 & 0 & {1.0000 + {0.0000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{15} = \begin{matrix}{0.9919 - {0.1270\; i}} & 0 & 0 & 0 \\0 & {0.9356 - {0.3532\; i}} & 0 & 0 \\0 & 0 & {0.9000 - {0.4359i}} & 0 \\0 & 0 & 0 & {1.0000 + {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{16} = \begin{matrix}{1.0000 + {0.0000\; i}} & 0 & 0 & 0 \\0 & {1.0000 - {0.0000\; i}} & 0 & 0 \\0 & 0 & {1.0000 - {0.0000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$

(7) Time correlation coefficient=0.95

${\overset{\sim}{\Theta}}_{1} = \begin{matrix}{0.9954 + {0.0960\; i}} & 0 & 0 & 0 \\0 & {0.9655 + {0.2604\; i}} & 0 & 0 \\0 & 0 & {0.9500 + {0.3122i}} & 0 \\0 & 0 & 0 & {1.0000 + {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{2} = \begin{matrix}{0.9827 + {0.1853\; i}} & 0 & 0 & 0 \\0 & {0.9571 + {0.2898\; i}} & 0 & 0 \\0 & 0 & {1.0000 + {0.0000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{3} = \begin{matrix}{0.9655 + {0.2604i}} & 0 & 0 & 0 \\0 & {0.9841 - {0.1778\; i}} & 0 & 0 \\0 & 0 & {0.9500 - {0.3122\; i}} & 0 \\0 & 0 & 0 & {1.0000 + {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{4} = \begin{matrix}{0.9500 + {0.3122\; i}} & 0 & 0 & 0 \\0 & {0.9500 - {0.3122\; i}} & 0 & 0 \\0 & 0 & {1.0000 - {0.0000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{5} = \begin{matrix}{0.9446 + {0.3281\; i}} & 0 & 0 & 0 \\0 & {0.9954 - {0.0960\; i}} & 0 & 0 \\0 & 0 & {0.9500 + {0.3122\; i}} & 0 \\0 & 0 & 0 & {1.0000 + {0.0000\; i}}\end{matrix}$

${\overset{\sim}{\Theta}}_{6} = \begin{matrix}{0.9571 + {0.2898i}} & 0 & 0 & 0 \\0 & {0.9827 + {0.1853\; i}} & 0 & 0 \\0 & 0 & {1.0000 + {0.0000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{7} = \begin{matrix}{0.9841 + {0.1778i}} & 0 & 0 & 0 \\0 & {0.9446 + {0.3281\; i}} & 0 & 0 \\0 & 0 & {0.9500 - {0.3122\; i}} & 0 \\0 & 0 & 0 & {1.0000 + {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{8} = \begin{matrix}{1.0000 - {0.0000\; i}} & 0 & 0 & 0 \\0 & {1.0000 + {0.0000\; i}} & 0 & 0 \\0 & 0 & {1.0000 - {0.0000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{9} = \begin{matrix}{0.9841 - {0.1778i}} & 0 & 0 & 0 \\0 & {0.9446 - {0.3281\; i}} & 0 & 0 \\0 & 0 & {0.9500 + {0.3122\; i}} & 0 \\0 & 0 & 0 & {1.0000 + {0.0000\; i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{10} = \begin{matrix}{0.9571 - {0.2898i}} & 0 & 0 & 0 \\0 & {0.9827 - {0.1853i}} & 0 & 0 \\0 & 0 & {1.0000 + {0.0000\; i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000\; i}}\end{matrix}$

${\overset{\sim}{\Theta}}_{11} = \begin{matrix}{0.9446 - {0.3281i}} & 0 & 0 & 0 \\0 & {0.9954 + {0.0960i}} & 0 & 0 \\0 & 0 & {0.9500 - {0.3122i}} & 0 \\0 & 0 & 0 & {1.0000 + {0.0000i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{12} = \begin{matrix}{0.9500 - {0.3122i}} & 0 & 0 & 0 \\0 & {0.9500 + {0.3122i}} & 0 & 0 \\0 & 0 & {1.0000 - {0.0000i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{13} = \begin{matrix}{0.9655 - {0.2604i}} & 0 & 0 & 0 \\0 & {0.9841 + {0.1778i}} & 0 & 0 \\0 & 0 & {0.9500 + {0.3122i}} & 0 \\0 & 0 & 0 & {1.0000 + {0.0000i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{14} = \begin{matrix}{0.9827 - {0.1853i}} & 0 & 0 & 0 \\0 & {0.9571 - {0.2898i}} & 0 & 0 \\0 & 0 & {1.0000 + {0.0000i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{15} = \begin{matrix}{0.9954 - {0.0960i}} & 0 & 0 & 0 \\0 & {0.9655 - {0.2604i}} & 0 & 0 \\0 & 0 & {0.9500 - {0.3122i}} & 0 \\0 & 0 & 0 & {1.0000 + {0.0000i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{16} = \begin{matrix}{1.0000 + {0.0000i}} & 0 & 0 & 0 \\0 & {1.0000 - {0.0000i}} & 0 & 0 \\0 & 0 & {1.0000 - {0.0000i}} & 0 \\0 & 0 & 0 & {1.0000 - {0.0000i}}\end{matrix}$

3) Where N_(t)=2, the number of feedback bits B=3, {θ}={Θ₁, . . . , Θ₂_(B) } includes full unitary matrices, and the first scheme, that is,the above Equation 14 or the above Equation 15 is used, examples of thecodebook {{tilde over (θ)}}={{tilde over (Θ)}₁, . . . , {tilde over(Θ)}₂ _(B) } that is updated according to the time correlationcoefficient may follow as:

(1) Time correlation coefficient=0

${\overset{\sim}{\Theta}}_{1} = \begin{matrix}{{- 0.0136} + {0.6753i}} & {0.7368 - {0.0288i}} \\{{- 0.7276} + {0.1198i}} & {{- 0.1489} - {0.6588i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{2} = \begin{matrix}{{- 0.6021} + {0.6871i}} & {0.0081 + {0.4065i}} \\{{- 0.0729} + {0.4000i}} & {{- 0.4847} - {0.7744i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{3} = \begin{matrix}{{- 0.0877} - {0.9095i}} & {0.3790 + {0.1464i}} \\{{- 0.3929} - {0.1035i}} & {{- 0.6041} + {0.6856i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{4} = \begin{matrix}{{- 0.7424} + {0.3706i}} & {0.2022 - {0.5202i}} \\{0.3856 - {0.4035i}} & {0.0213 - {0.8295i}}\end{matrix}$

${\overset{\sim}{\Theta}}_{5} = \begin{matrix}{{- 0.2839} + {0.0675i}} & {0.7744 + {0.5614i}} \\{0.6104 + {0.7363i}} & {{- 0.2582} - {0.1359i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{6} = \begin{matrix}{{- 0.4786} + {0.1916i}} & {0.8547 + {0.0614i}} \\{{- 0.0827} + {0.8529i}} & {{- 0.2692} + {0.4397i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{7} = \begin{matrix}{{- 0.1309} - {0.8846i}} & {{- 0.4081} - {0.1838i}} \\{0.1516 + {0.4211i}} & {{- 0.8718} - {0.1992i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{8} = \begin{matrix}{{- 0.0707} + {0.9650i}} & {{- 0.0601} - {0.2455i}} \\{{- 0.1422} + {0.2090i}} & {{- 0.2711} + {0.9288i}}\end{matrix}$

(2) Time correlation coefficient=0.7

${\overset{\sim}{\Theta}}_{1} = \begin{matrix}{0.6945 + {0.4645i}} & {0.5495 + {0.0057i}} \\{{- 0.5460} + {0.0624i}} & {0.6429 - {0.5336i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{2} = \begin{matrix}{0.4276 + {0.7949i}} & {0.0251 + {0.4298i}} \\{{- 0.0609} + {0.4262i}} & {0.4926 - {0.7563i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{3} = \begin{matrix}{0.6647 - {0.6524i}} & {0.3554 + {0.0784i}} \\{{- 0.3619} - {0.0386i}} & {0.4228 + {0.8299i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{4} = \begin{matrix}{0.2975 + {0.7951i}} & {0.0529 - {0.5259i}} \\{0.2498 - {0.4658i}} & {0.6908 - {0.4934i}}\end{matrix}$

${\overset{\sim}{\Theta}}_{5} = \begin{matrix}{0.5853 + {0.1239i}} & {{- 0.6188} + {0.5091i}} \\{0.5481 + {0.5845i}} & {0.5964 - {0.0475i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{6} = \begin{matrix}{0.3849 - {0.3268i}} & {0.7954 + {0.3352i}} \\{{- 0.3550} + {0.7868i}} & {0.5044 - {0.0223i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{7} = \begin{matrix}{0.5859 - {0.7498i}} & {{- 0.0341} - {0.3057i}} \\{{- 0.1961} + {0.2370i}} & {{- 0.0789} - {0.9482i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{8} = \begin{matrix}{0.6661 + {0.7343i}} & {{- 0.1167} - {0.0590i}} \\{0.0342 + {0.1262i}} & {0.5844 + {0.8009i}}\end{matrix}$

(3) Time correlation coefficient=0.75

${\overset{\sim}{\Theta}}_{1} = \begin{matrix}{0.7458 + {0.4303i}} & {0.5085 + {0.0094i}} \\{{- 0.5057} + {0.0537i}} & {0.7054 - {0.4937i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{2} = \begin{matrix}{0.5543 + {0.7317i}} & {0.0282 + {0.3957i}} \\{{- 0.0511} + {0.3934i}} & {0.5959 - {0.6982i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{3} = \begin{matrix}{0.7233 - {0.6044i}} & {0.3287 + {0.0596i}} \\{{- 0.3332} - {0.0229i}} & {0.5528 + {0.7635i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{4} = \begin{matrix}{0.5082 + {0.7201i}} & {0.0038 - {0.4725i}} \\{0.1840 - {0.4352i}} & {0.7523 - {0.4592i}}\end{matrix}$

${\overset{\sim}{\Theta}}_{5} = \begin{matrix}{0.6624 + {0.1139i}} & {{- 0.5658} + {0.4776i}} \\{0.5132 + {0.5337i}} & {0.6707 - {0.0446i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{6} = \begin{matrix}{0.6176 - {0.2742i}} & {0.6233 + {0.3935i}} \\{{- 0.4090} + {0.6132i}} & {0.6758 - {0.0011i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{7} = \begin{matrix}{0.7615 - {0.6455i}} & {{- 0.0299} + {0.0507i}} \\{0.0573 - {0.0137i}} & {0.8405 - {0.5387i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{8} = \begin{matrix}{0.7257 + {0.6795i}} & {{- 0.1001} - {0.0397i}} \\{0.0375 + {0.1009i}} & {0.6637 + {0.7402i}}\end{matrix}$

(4) Time correlation coefficient=0.8

${\overset{\sim}{\Theta}}_{1} = \begin{matrix}{0.7970 + {0.3909i}} & {0.4602 + {0.0124i}} \\{{- 0.4582} + {0.0447i}} & {0.7676 - {0.4459i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{2} = \begin{matrix}{0.6756 + {0.6474i}} & {0.0295 + {0.3516i}} \\{{- 0.0410} + {0.3504i}} & {0.6965 - {0.6248i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{3} = \begin{matrix}{0.7814 - {0.5487i}} & {0.2943 + {0.0417i}} \\{{- 0.2971} - {0.0089i}} & {0.6767 + {0.6736i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{4} = \begin{matrix}{0.6909 + {0.5994i}} & {{- 0.0321} - {0.4029i}} \\{0.1242 - {0.3846i}} & {0.8105 - {0.4238i}}\end{matrix}$

${\overset{\sim}{\Theta}}_{5} = \begin{matrix}{0.7385 + {0.1002i}} & {{- 0.5038} + {0.4367i}} \\{0.4685 + {0.4745i}} & {0.7440 - {0.0426i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{6} = \begin{matrix}{0.7863 - {0.1721i}} & {0.4547 + {0.3811i}} \\{{- 0.3923} + {0.4451i}} & {0.8033 + {0.0521i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{7} = \begin{matrix}{0.8203 - {0.5564i}} & {{- 0.0878} + {0.0991i}} \\{0.1323 - {0.0058i}} & {0.9488 - {0.2868i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{8} = \begin{matrix}{0.7847 + {0.6142i}} & {{- 0.0801} - {0.0229i}} \\{0.0366 + {0.0749i}} & {0.7416 + {0.6657i}}\end{matrix}$

(5) Time correlation coefficient=0.85

${\overset{\sim}{\Theta}}_{1} = \begin{matrix}{0.8482 + {0.3441i}} & {0.4025 + {0.0144i}} \\{{- 0.4012} + {0.0355i}} & {0.8290 - {0.3881i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{2} = \begin{matrix}{0.7852 + {0.5424i}} & {0.0283 + {0.2973i}} \\{{- 0.0314} + {0.2970i}} & {0.7907 - {0.5344i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{3} = \begin{matrix}{0.8386 - {0.4823i}} & {0.2519 + {0.0263i}} \\{{- 0.2532} + {0.0017i}} & {0.7877 + {0.5616i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{4} = \begin{matrix}{0.8263 + {0.4568i}} & {{- 0.0498} - {0.3256i}} \\{0.0784 - {0.3200i}} & {0.8635 - {0.3820i}}\end{matrix}$

${\overset{\sim}{\Theta}}_{5} = \begin{matrix}{0.8121 + {0.0827i}} & {{- 0.4314} + {0.3840i}} \\{0.4112 + {0.4056i}} & {0.8153 - {0.0411i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{6} = \begin{matrix}{0.8840 - {0.0781i}} & {0.3266 + {0.3253i}} \\{{- 0.3334} + {0.3184i}} & {0.8821 + {0.0966i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{7} = \begin{matrix}{0.8670 - {0.4787i}} & {{- 0.1019} + {0.0943i}} \\{0.1386 + {0.0077i}} & {0.9730 - {0.1841i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{8} = \begin{matrix}{0.8424 + {0.5356i}} & {{- 0.0576} - {0.0100i}} \\{0.0311 + {0.0495i}} & {0.8162 + {0.5748i}}\end{matrix}$

(6) Time correlation coefficient=0.9

${\overset{\sim}{\Theta}}_{1} = \begin{matrix}{0.8991 + {0.2860i}} & {0.3310 + {0.0149i}} \\{{- 0.3303} + {0.0262i}} & {0.8889 - {0.3164i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{2} = \begin{matrix}{0.8780 + {0.4179i}} & {0.0241 + {0.2322i}} \\{{- 0.0225} + {0.2323i}} & {0.8751 - {0.4239i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{3} = \begin{matrix}{0.8945 - {0.3994i}} & {0.2004 + {0.0145i}} \\{{- 0.2008} + {0.0077i}} & {0.8804 + {0.4295i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{4} = \begin{matrix}{0.9156 + {0.3141i}} & {{- 0.0505} - {0.2460i}} \\{0.0473 - {0.2466i}} & {0.9114 - {0.3260i}}\end{matrix}$

${\overset{\sim}{\Theta}}_{5} = \begin{matrix}{0.8817 + {0.0616i}} & {{- 0.3454} + {0.3154i}} \\{0.3371 + {0.3243i}} & {0.8830 - {0.0386i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{6} = \begin{matrix}{0.9394 - {0.0135i}} & {0.2318 + {0.2523i}} \\{{- 0.2580} + {0.2254i}} & {0.9323 + {0.1162i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{7} = \begin{matrix}{0.9118 - {0.3918i}} & {{- 0.0969} + {0.0762i}} \\{0.1222 + {0.0167i}} & {0.9847 - {0.1234i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{8} = \begin{matrix}{0.8980 + {0.4387i}} & {{- 0.0340} - {0.0017i}} \\{0.0215 + {0.0264i}} & {0.8856 + {0.4632i}}\end{matrix}$

(7) Time correlation coefficient=0.95

${\overset{\sim}{\Theta}}_{1} = \begin{matrix}{0.9498 + {0.2064i}} & {0.2347 + {0.0128i}} \\{{- 0.2345} + {0.0164i}} & {0.9465 - {0.2211i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{2} = \begin{matrix}{0.9505 + {0.2699i}} & {0.0165 + {0.1528i}} \\{{- 0.0142} + {0.1530i}} & {0.9463 - {0.2844i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{3} = \begin{matrix}{0.9485 - {0.2861i}} & {0.1357 + {0.0067i}} \\{{- 0.1357} + {0.0083i}} & {0.9519 + {0.2747i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{4} = \begin{matrix}{0.9699 + {0.1789i}} & {{- 0.0374} - {0.1609i}} \\{0.0269 - {0.1630i}} & {0.9561 - {0.2419i}}\end{matrix}$

${\overset{\sim}{\Theta}}_{5} = \begin{matrix}{0.9454 + {0.0369i}} & {{- 0.2368} + {0.2209i}} \\{0.2358 + {0.2220i}} & {0.9456 - {0.0325i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{6} = \begin{matrix}{0.9744 + {0.0209i}} & {0.1499 + {0.1665i}} \\{{- 0.1702} + {0.1457i}} & {0.9688 + {0.1056i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{7} = \begin{matrix}{0.9560 - {0.2786i}} & {{- 0.0767} + {0.0503i}} \\{0.0894 + {0.0201i}} & {0.9929 - {0.0752i}}\end{matrix}$ ${\overset{\sim}{\Theta}}_{8} = \begin{matrix}{0.9508 + {0.3097i}} & {{- 0.0117} + {0.0012i}} \\{0.0087 + {0.0079i}} & {0.9478 + {0.3187i}}\end{matrix}$

In addition to the aforementioned examples, {{tilde over (θ)}}={{tildeover (Θ)}₁, . . . , {tilde over (Θ)}₂ _(B) } may be variously calculatedaccording to various time coefficients, the number of feedback bits, thenumber of transmit antennas of the base station, or {θ}={Θ₁, . . . , Θ₂_(B) }. Further descriptions related thereto will be omitted herein forconciseness.

FIG. 3 is a flowchart illustrating a MIMO communication method accordingto an exemplary embodiment. It is understood that one or moreapparatuses described above, for example, a base station, may carry outone or more of the operations of FIG. 3.

Referring to FIG. 3, a pre-designed codebook may be stored, for example,in a memory, in operation S310. The exemplary design scheme of thecodebook has been described above. A base station or a user terminal maystore and use the same codebook.

A state of a channel that is formed between the base station and theuser terminal may be recognized in operation S320.

For example, the base station may transmit a well-known pilot signal tothe user terminal and the user terminal may estimate a channel formedbetween the base station and the user terminal using the pilot signal. Achannel state of the estimated channel may be expressed as channelinformation. The channel information may include channel stateinformation, channel quality information, or channel directioninformation. The channel information may be fed back from the userterminal to the base station. The base station may recognize the channelstate based on the fed back channel information.

A transmission rank may be adaptively determined in operation S330. Forexample, the transmission rank may be adaptively determined according toan achievable total data transmission rate, a state of channels ofusers/user terminals, or data desired by the users/user terminals.

In operation S340, a precoding matrix may be determined by consideringthe channel information and the transmission rank based on a pluralityof matrices that is included in the codebook. For example, a matrix maybe selected from the plurality of matrices based on the channelinformation. The size of the selected matrix may be adjusted accordingto the transmission rank and thereby be used as the precoding matrix.

Data streams may be precoded using the determined precoding matrix inoperation S350.

FIG. 4 is a flowchart illustrating a MIMO communication method accordingto another embodiment. It is understood that one or more apparatusesdescribed above, for example, a base station, may carry out one or moreof the operations of FIG. 4.

Referring to FIG. 4, a pre-designed codebook may be stored, for example,in a memory, in operation S410.

A channel state of a channel that is formed between a user and a basestation may be recognized in operation S420. For example, a base stationmay recognize the channel state based on channel information that is fedback from a user terminal.

A time correlation coefficient of the channel may be recognized inoperation S430. The user terminal may calculate the time correlationcoefficient of the channel and quantize the calculated time correlationcoefficient. The user terminal may feed back the quantized value to thebase station.

The codebook may be updated according to a time correlation coefficient(p) of a channel that is formed between at least one user and the basestation, in operation S440. Schemes of updating the codebook have beendescribed above and thus further descriptions related thereto will beomitted.

A precoding matrix may be generated using the updated codebook inoperation S450. For example, where the updated codebook is {{tilde over(θ)}}={{tilde over (Θ)}₁, . . . , {tilde over (Θ)}₂ _(B) }, and apreviously used precoding matrix is F_(τ-1), a currently used precodingmatrix may be generated using F_(τ)={tilde over (Θ)}_(i)F_(τ-1).

Data streams may be precoded using the generated precoding matrix inoperation S460.

According to certain embodiments described above, it is possible toimprove a data transmission rate using a codebook that is optimizedaccording to a channel environment, a transmission rank, and/or a numberof feedback bits.

The methods described above including a MIMO communication method may berecorded, stored, or fixed in one or more computer-readable media thatincludes program instructions to be implemented by a computer to cause aprocessor to execute or perform the program instructions. The media mayalso include, alone or in combination with the program instructions,data files, data structures, and the like. The media and programinstructions may be those specially designed and constructed, or theymay be of the kind well-known and available to those having skill in thecomputer software arts. Examples of computer-readable media includemagnetic media such as hard disks, floppy disks, and magnetic tape;optical media such as CD ROM disks and DVD; magneto-optical media suchas optical disks; and hardware devices that are specially configured tostore and perform program instructions, such as read-only memory (ROM),random access memory (RAM), flash memory, and the like. Examples ofprogram instructions include both machine code, such as produced by acompiler, and files containing higher level code that may be executed bythe computer using an interpreter. The described hardware devices may beconfigured to act as one or more software modules in order to performthe operations and methods described above, or vice versa. In addition,a computer-readable storage or recording medium may be distributed amongcomputer systems connected through a network and computer-readableinstructions or codes may be stored and executed in a decentralizedmanner.

A computer or a computing system may include a microprocessor that iselectrically connected with a bus, a user interface, a modem such as abaseband chipset, a memory controller, and a flash memory device. Theflash memory device may store N-bit data via the memory controller. TheN-bit data is processed or will be processed by the microprocessor and Nmay be 1 or an integer greater than 1. Where the computer or thecomputing system is a mobile apparatus, a battery may be additionallyprovided to supply operation voltage of the computer of the computingsystem. It will be apparent to those of ordinary skill in the art thatthe computer or the computing system may further include an applicationchipset, a camera image processor (CIS), a mobile Dynamic Random AccessMemory (DRAM), and the like. The memory controller and the flash memorydevice may constitute a solid state drive/disk (SSD) that uses anon-volatile memory to store data.

A number of exemplary embodiments have been described above.Nevertheless, it will be understood that various modifications may bemade. For example, suitable results may be achieved if the describedtechniques are performed in a different order and/or if components in adescribed system, architecture, device, or circuit are combined in adifferent manner and/or replaced or supplemented by other components ortheir equivalents. Accordingly, other implementations are within thescope of the following claims.

What is claimed is:
 1. A base station for a single user multiple inputmultiple output (MIMO) communication system, the base stationcomprising: a memory where a codebook including codeword matricesC_(1,1), C_(2,1), C_(3,1), C_(4,1), C_(5,1), C_(6,1), C_(7,1), C_(8,1),C_(9,1), C_(10,1), C_(11,1), C_(12,1), C_(13,1), C_(14,1), C_(15,1), andC_(16,1) is stored; and a precoder to precode a data stream to betransmitted using the codebook, wherein the codeword matrices aredefined by the following table: C_(1,1) = C_(2,1) = C_(3,1) = C_(4,1) =  0.5000 −0.5000 −0.5000   0.5000 −0.5000 −0.5000   0.5000 0 − 0.5000i  0.5000   0.5000   0.5000   0.5000 −0.5000   0.5000 −0.5000 0 − 0.5000iC_(5,1) = C_(6,1) = C_(7,1) = C_(8,1) = −0.5000 −0.5000   0.5000  0.5000 0 − 0.5000i 0 + 0.5000i   0.5000 0 + 0.5000i   0.5000   0.5000  0.5000   0.5000 0 + 0.5000i 0 − 0.5000i   0.5000 0 + 0.5000i C_(9,1) =C_(10,1) = C_(11,1) = C_(12,1) =   0.5000   0.5000   0.5000   0.5000  0.5000 0 + 0.5000i −0.5000 0 − 0.5000i   0.5000 −0.5000   0.5000−0.5000 −0.5000 0 + 0.5000i   0.5000 0 − 0.5000i C_(13,1) = C_(14,1) =C_(15,1) = C_(16,1) =   0.5000   0.5000   0.5000   0.5000 0.3536 +0.3536i −0.3536 + −0.3536 − 0.3536 − 0.3536i    0.3536i   0.3536i 0 +0.5000i 0 − 0.5000i 0 + 0.5000i 0 − 0.5000i −0.3536 + 0.3536 + 0.3536i0.3536 − 0.3536i −0.3536 −   0.3536i   0.3536i.


2. The base station of claim 1, wherein the precoder calculates aprecoding matrix based on at least one codeword matrix among thecodeword matrices, and precodes the data stream using the precodingmatrix.
 3. The base station of claim 1, further comprising: aninformation receiver to receive feedback information from a terminal,wherein the precoder precodes the data stream using the feedbackinformation and the codebook.
 4. The base station of claim 3, whereinthe precoder calculates a precoding matrix based on a codeword matrixcorresponding to the feedback information among the codeword matrices,and precodes the data stream using the precoding matrix.
 5. The basestation of claim 3, wherein the feedback information includes indexinformation of a codeword matrix preferred by the terminal among thecodeword matrices.
 6. The base station of claim 1, further comprisingfour transmit antennas, wherein the codebook is used for transmissionrank
 1. 7. A base station for a multi-user MIMO communication system,the base station comprising: a memory where a codebook includingcodeword matrices C_(1,1), C_(2,1), C_(3,1), C_(4,1), C_(5,1), C_(6,1),C_(7,1), C_(8,1), C_(9,1), C_(10,1), C_(11,1), C_(12,1), C_(13,1),C_(14,1), C_(15,1), and C_(16,1) is stored; and a precoder to precode atleast one data stream to be transmitted using the codebook, wherein thecodeword matrices are defined by the following table: C_(1,1) = C_(2,1)= C_(3,1) = C_(4,1) =   0.5000 −0.5000 −0.5000   0.5000 −0.5000 −0.5000  0.5000 0 − 0.5000i   0.5000   0.5000   0.5000   0.5000 −0.5000  0.5000 −0.5000 0 − 0.5000i C_(5,1) = C_(6,1) = C_(7,1) = C_(8,1) =−0.5000 −0.5000   0.5000   0.5000 0 − 0.5000i 0 + 0.5000i   0.5000 0 +0.5000i   0.5000   0.5000   0.5000   0.5000 0 + 0.5000i 0 − 0.5000i  0.5000 0 + 0.5000i C_(9,1) = C_(10,1) = C_(11,1) = C_(12,1) =   0.5000  0.5000   0.5000   0.5000   0.5000 0 + 0.5000i −0.5000 0 − 0.5000i  0.5000 −0.5000   0.5000 −0.5000 −0.5000 0 + 0.5000i   0.5000 0 −0.5000i C_(13,1) = C_(14,1) = C_(15,1) = C_(16,1) =   0.5000   0.5000  0.5000   0.5000 0.3536 + 0.3536i −0.3536 + −0.3536 − 0.3536 − 0.3536i   0.3536i   0.3536i 0 + 0.5000i 0 − 0.5000i 0 + 0.5000i 0 − 0.5000i−0.3536 + 0.3536 + 0.3536i 0.3536 − 0.3536i −0.3536 −   0.3536i  0.3536i.


8. The base station of claim 7, further comprising: an informationreceiver to receive feedback information from at least two terminals,wherein the precoder precodes the at least one data stream using atleast one of the feedback information received from the at least twoterminals, and the codebook.
 9. A terminal for a MIMO communicationsystem, the terminal comprising: a memory where a codebook includingcodeword matrices C_(1,1), C_(2,1), C_(3,1), C_(4,1), C_(5,1), C_(6,1),C_(7,1), C_(8,1), C_(9,1), C_(10,1), C_(11,1), C_(12,1), C_(13,1),C_(14,1), C_(15,1), and C_(16,1) is stored; and a feedback unit to feedback, to a base station, feedback information associated with apreferred codeword matrix among the codeword matrices, wherein thecodeword matrices are defined by the following table: C_(1,1) = C_(2,1)= C_(3,1) = C_(4,1) =   0.5000 −0.5000 −0.5000   0.5000 −0.5000 −0.5000  0.5000 0 − 0.5000i   0.5000   0.5000   0.5000   0.5000 −0.5000  0.5000 −0.5000 0 − 0.5000i C_(5,1) = C_(6,1) = C_(7,1) = C_(8,1) =−0.5000 −0.5000   0.5000   0.5000 0 − 0.5000i 0 + 0.5000i   0.5000 0 +0.5000i   0.5000   0.5000   0.5000   0.5000 0 + 0.5000i 0 − 0.5000i  0.5000 0 + 0.5000i C_(9,1) = C_(10,1) = C_(11,1) = C_(12,1) =   0.5000  0.5000   0.5000   0.5000   0.5000 0 + 0.5000i −0.5000 0 − 0.5000i  0.5000 −0.5000   0.5000 −0.5000 −0.5000 0 + 0.5000i   0.5000 0 −0.5000i C_(13,1) = C_(14,1) = C_(15,1) = C_(16,1) =   0.5000   0.5000  0.5000   0.5000 0.3536 + 0.3536i −0.3536 + −0.3536 − 0.3536 − 0.3536i   0.3536i   0.3536i 0 + 0.5000i 0 − 0.5000i 0 + 0.5000i 0 − 0.5000i−0.3536 + 0.3536 + 0.3536i 0.3536 − 0.3536i −0.3536 −   0.3536i  0.3536i.


10. The terminal of claim 9, further comprising: a channel estimationunit to estimate a channel between the base station and the terminal,wherein the feedback unit feeds back, to the base station, the feedbackinformation determined based on the estimated channel.
 11. Anon-transitory information storage medium having stored therein acodebook used by a base station and at least one terminal of a MIMOcommunication system, wherein the codebook includes codeword matricesC_(1,1), C_(2,1), C_(3,1), C_(4,1), C_(5,1), C_(6,1), C_(7,1), C_(8,1),C_(9,1), C_(10,1), C_(11,1), C_(12,1), C_(13,1), C_(14,1), C_(15,1), andC_(16,1), and the codeword matrices are defined by the following table:C_(1,1) =   0.5000 C_(2,1) = −0.5000 C_(3,1) = −0.5000 C_(4,1) =  0.5000 −0.5000 −0.5000   0.5000 0 − 0.5000i   0.5000   0.5000   0.5000  0.5000 −0.5000   0.5000 −0.5000 0 − 0.5000i C_(5,1) = −0.5000 C_(6,1)= −0.5000 C_(7,1) =   0.5000 C_(8,1) =   0.5000 0 − 0.5000i 0 + 0.5000i  0.5000 0 + 0.5000i   0.5000   0.5000   0.5000   0.5000 0 + 0.5000i 0 −0.5000i   0.5000 0 + 0.5000i C_(9,1) =   0.5000 C_(10,1) =   0.5000C_(11,1) =   0.5000 C_(12,1) =   0.5000   0.5000 0 + 0.5000i −0.5000 0 −0.5000i   0.5000 −0.5000   0.5000 −0.5000 −0.5000 0 + 0.5000i   0.5000 0− 0.5000i C_(13,1) =   0.5000 C_(14,1) =   0.5000 C_(15,1) =   0.5000C_(16,1) =   0.5000 0.3536 + 0.3536i −0.3536 + 0.3536i −0.3536 − 0.3536i0.3536 − 0.3536i 0 + 0.5000i 0 − 0.5000i 0 + 0.5000i 0 − 0.5000i−0.3536 + 0.3536i 0.3536 + 0.3536i 0.3536 − 0.3536i −0.3536 − 0.3536i.


12. A precoding method of a base station for a single user MIMOcommunication system, the method comprising: accessing a memory where acodebook including codeword matrices C_(1,1), C_(2,1), C_(3,1), C_(4,1),C_(5,1), C_(6,1), C_(7,1), C_(8,1), C_(9,1), C_(10,1), C_(11,1),C_(12,1), C_(13,1), C_(14,1), C_(15,1), and C_(16,1) is stored; andprecoding a data stream to be transmitted using the codebook, whereinthe codeword matrices are defined by the following table: C_(1,1) =  0.5000 C_(2,1) = −0.5000 C_(3,1) = −0.5000 C_(4,1) =   0.5000 −0.5000−0.5000   0.5000 0 − 0.5000i   0.5000   0.5000   0.5000   0.5000 −0.5000  0.5000 −0.5000 0 − 0.5000i C_(5,1) = −0.5000 C_(6,1) = −0.5000 C_(7,1)=   0.5000 C_(8,1) =   0.5000 0 − 0.5000i 0 + 0.5000i   0.5000 0 +0.5000i   0.5000   0.5000   0.5000   0.5000 0 + 0.5000i 0 − 0.5000i  0.5000 0 + 0.5000i C_(9,1) =   0.5000 C_(10,1) =   0.5000 C_(11,1) =  0.5000 C_(12,1) =   0.5000   0.5000 0 + 0.5000i −0.5000 0 − 0.5000i  0.5000 −0.5000   0.5000 −0.5000 −0.5000 0 + 0.5000i   0.5000 0 −0.5000i C_(13,1) =   0.5000 C_(14,1) =   0.5000 C_(15,1) =   0.5000C_(16,1) =   0.5000 0.3536 + 0.3536i −0.3536 + 0.3536i −0.3536 − 0.3536i0.3536 − 0.3536i 0 + 0.5000i 0 − 0.5000i 0 + 0.5000i 0 − 0.5000i−0.3536 + 0.3536i 0.3536 + 0.3536i 0.3536 − 0.3536i −0.3536 − 0.3536i.


13. The method of claim 12, wherein the precoding comprises: calculatinga precoding matrix based on at least one codeword matrix among thecodeword matrices; and precoding the data stream using the precodingmatrix.
 14. The method of claim 12, further comprising: receivingfeedback information from a terminal, wherein the precoding comprisesprecoding the data stream using the feedback information and thecodebook.
 15. A precoding method of a base station for a multi-user MIMOcommunication system, the method comprising: accessing a memory where acodebook including codeword matrices C_(1,1), C_(2,1), C_(3,1), C_(4,1),C_(5,1), C_(6,1), C_(7,1), C_(8,1), C_(9,1), C_(10,1), C_(11,1),C_(12,1), C_(13,1), C_(14,1), C_(15,1), and C_(16,1) is stored; andprecoding at least one data stream to be transmitted using the codebook,wherein the codeword matrices are defined by the following table:C_(1,1) =   0.5000 C_(2,1) = −0.5000 C_(3,1) = −0.5000 C_(4,1) =  0.5000 −0.5000 −0.5000   0.5000 0 − 0.5000i   0.5000   0.5000   0.5000  0.5000 −0.5000   0.5000 −0.5000 0 − 0.5000i C_(5,1) = −0.5000 C_(6,1)= −0.5000 C_(7,1) =   0.5000 C_(8,1) =   0.5000 0 − 0.5000i 0 + 0.5000i  0.5000 0 + 0.5000i   0.5000   0.5000   0.5000   0.5000 0 + 0.5000i 0 −0.5000i   0.5000 0 + 0.5000i C_(9,1) =   0.5000 C_(10,1) =   0.5000C_(11,1) =   0.5000 C_(12,1) =   0.5000   0.5000 0 + 0.5000i −0.5000 0 −0.5000i   0.5000 −0.5000   0.5000 −0.5000 −0.5000 0 + 0.5000i   0.5000 0− 0.5000i C_(13,1) =   0.5000 C_(14,1) =   0.5000 C_(15,1) =   0.5000C_(16,1) =   0.5000 0.3536 + 0.3536i −0.3536 + 0.3536i −0.3536 − 0.3536i0.3536 − 0.3536i 0 + 0.5000i 0 − 0.5000i 0 + 0.5000i 0 − 0.5000i−0.3536 + 0.3536i 0.3536 + 0.3536i 0.3536 − 0.3536i −0.3536 − 0.3536i.


16. A method of a terminal for a MIMO communication system, the methodcomprising: accessing a memory where a codebook including codewordmatrices C_(1,1), C_(2,1), C_(3,1), C_(4,1), C_(5,1), C_(6,1), C_(7,1),C_(8,1), C_(9,1), C_(10,1), C_(11,1), C_(12,1), C_(13,1), C_(14,1),C_(15,1), and C_(16,1) is stored; and feeding back, to a base station,feedback information associated with a preferred codeword matrix amongthe codeword matrices, wherein the codeword matrices are defined by thefollowing table: C_(1,1) =   0.5000 C_(2,1) = −0.5000 C_(3,1) = −0.5000C_(4,1) =   0.5000 −0.5000 −0.5000   0.5000 0 − 0.5000i   0.5000  0.5000   0.5000   0.5000 −0.5000   0.5000 −0.5000 0 − 0.5000i C_(5,1)= −0.5000 C_(6,1) = −0.5000 C_(7,1) =   0.5000 C_(8,1) =   0.5000 0 −0.5000i 0 + 0.5000i   0.5000 0 + 0.5000i   0.5000   0.5000   0.5000  0.5000 0 + 0.5000i 0 − 0.5000i   0.5000 0 + 0.5000i C_(9,1) =   0.5000C_(10,1) =   0.5000 C_(11,1) =   0.5000 C_(12,1) =   0.5000   0.5000 0 +0.5000i −0.5000 0 − 0.5000i   0.5000 −0.5000   0.5000 −0.5000 −0.50000 + 0.5000i   0.5000 0 − 0.5000i C_(13,1) =   0.5000 C_(14,1) =   0.5000C_(15,1) =   0.5000 C_(16,1) =   0.5000 0.3536 + 0.3536i −0.3536 +0.3536i −0.3536 − 0.3536i 0.3536 − 0.3536i 0 + 0.5000i 0 − 0.5000i 0 +0.5000i 0 − 0.5000i −0.3536 + 0.3536i 0.3536 + 0.3536i 0.3536 − 0.3536i−0.3536 − 0.3536i.


17. The method of claim 16, further comprising: estimating a channelbetween the base station and the terminal; and generating the feedbackinformation based on the estimated channel.